regularity for p harmonic equations with right hand side in orlicz zygmund classes abstract we establish regularity results for solutions of some degenerate elliptic pdes with right hand side in a suitable orlicz zygmund class the nonnegative function which measures the degree of degeneracy of the ellipticity bounds is assumed to be exponentially integrable we find that the scale of improved regularity is logarithmic and we indicate its exact dependence on the degree of the degeneracy of the problem

a discrete multiple scales analysis of a discrete version of the korteweg de vries equation abstract a more elaborate discrete multiple scales analysis than that used by newell in 1977 is performed on the zabusky kruskal discretization of the korteweg de vries kdv equation this eventually leads to a set of partial difference equations describing the modulational behavior of a small harmonic wave modulated by a slowly varying envelope in the case of certain modes of the carrier wave the multiple scales analysis breaks down indicating that in these cases the numerical solution deviates in behavior from that of the kdv equation numerical experiments are reported which confirm this

asymptotic behavior of the solution to the non isothermal phase field equation we consider the penrose fife phase field model thermodynamically consistent models of phase field type for the kinetics of phase transitions physica d 43 1990 44 62 with homogeneous neumann boundary condition to the nonlinear heat flux q 1 i e q 0 on the boundary where 0 is the temperature there is a unique h1 solution globally in time with the non empty connected compact limit set composed of stationary solutions and the linearized stable stationary solution is dynamically stable

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

bi sobolev mappings and elliptic equations in the plane abstract suppose that f u v is a homeomorphism in the plane of the sobolev class w loc 1 1 such that its inverse is of the same sobolev class we prove that u and v have the same set of critical points as an application we show that u and v are distributional solutions to the same non trivial degenerate elliptic equation in divergence form we study similar properties also in higher dimensions

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

trudinger s exponential integrability for riesz potentials of functions in generalized grand morrey spaces abstract our aim in this paper is to discuss trudinger s exponential integrability for riesz potentials of functions in generalized grand morrey spaces our result will imply the boundedness of the riesz potential operator from a grand morrey space to a morrey space

very weak solutions of degenerate parabolic systems with non standard p x t growth abstract we study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p x t laplacian system t u div d u p x t 2 d u div f p x t 2 f under natural assumptions on the exponent function p 0 t 2 we prove that any very weak solution u 0 t r n with d u p 1 e l 1 belongs to the natural energy space i e d u p l loc 1 provided e 0 is small enough

asymptotic behavior of the solution to the non isothermal phase separation abstract we consider the non isothermal phase separation models of the penrose fife type which were proposed in o penrose p c fife thermodynamically consistent models of phase field type for the kinetics of phase transitions physica d 43 1990 44 62 with homogeneous neumann boundary conditions on the nonlinear heat flux q u i e q n 0 on the boundary of a region which the material occupies here u represents the absolute temperature for this model we first show that there exists a unique solution globally in time moreover the limit set associated with the trajectory of the unique global solution is non empty connected and compact in some suitable space as well as being composed of solutions to the steady state problem for the stability of stationary solutions we show that the dynamically stable solutions to the steady state problem are characterized by linearized stable solutions to the elliptic problem with a non local term which is equivalent to our steady state problem

existence and uniqueness for nonlinear elliptic equations with lower order terms abstract we study the nonlinear dirichlet problem for the elliptic equation div a x u b x u div f in a regular domain r n n 2 u 0 on in hypothesis of lipschitz continuity and strong monotonicity of a we assume that the lower order term b x s verify b x s b x t e x s t for a e x and for any s t r where e is a non negative function in the lorentz space l n q n q without any control on the norm of e with f l p p 1 and q we obtain existence and uniqueness result for distributional solutions u w 1 p whenever p is close to two for q uniqueness results are obtained the main difficulty to solve the problem is due to noncoercivity of the vector field a x s a x b x s moreover no classical structure conditions are satisfied

on the operator l f f log f abstract the present note deals with the operator l h 1 r n d r n which takes a given element f of the hardy space to the function f log f in general this function need not be locally integrable nevertheless due to peculiar cancellations of large positive and negative terms in the integral f log f with c 0 r n we are able to give meaning to f log f as a schwartz distribution we find several alternatives for this interpretation of f log f

multiple scale analysis of discrete nonlinear partial difference equations the reduction of the lattice potential kdv we consider multiple lattices and functions defined on them we introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices we use these results to perform multiple scale reduction of the lattice potential korteweg de vries equation

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

nonlinear schrodinger type equations from multiscale reduction of pdes ii necessary conditions of integrability for real pdes the multiscale method has been applied in j math phys 41 6399 2000 to evolution pdes to obtain various nonlinear schrodinger type equations here these results are exploited to derive necessary conditions of integrability for nonlinear real evolution pdes in 1 1 dimensions these conditions are given by explicit formulas and several examples are detailed

local regularity for solutions of anisotropic obstacle problems abstract this paper deals with anisotropic obstacle problems for the a harmonic equation div a x u x 0 a local regularity result is given under suitable assumptions

c integrable nonlinear pdes iv a technique based on a change of independent variables used in a previous paper to generate c integrable nonlinear pdes i e nonlinear pdes solvable by an appropriate change of variables in 1 1 dimensions one time and one space variable is extended to the case of more than one space dimension several examples of evolution c integrable pdes in 1 2 dimensions one time and two space variables are exhibited

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

c integrable nonlinear pdes ii a technique to perform a convenient change of independent variables in a pde is reported and it is used to generate c integrable nonlinear pdes i e nonlinear pdes solvable by an appropriate change of variables several examples of such pdes are exhibited

existence results for some nonuniformly elliptic equations with irregular data abstract in this paper we prove the existence of solutions of some nonlinear elliptic equations with degenerate coercivity of the type formula when the datum f belongs to the space lr where r 2 and 0 1

secondary bifurcation for a nonlocal allen cahn equation abstract this paper studies the neumann problem of a nonlocal allen cahn equation in an interval a main result finds a symmetry breaking secondary bifurcation point on the bifurcation curve of solutions with odd symmetry our proof is based on a level set analysis for the associated integral map a method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point we also show some numerical simulations concerning the global bifurcation structure

global higher integrability for the gradient of very weak solutions of a class of nonlinear elliptic systems abstract the global higher integrability for the gradient of very weak solutions of nonlinear elliptic systems div a x u du b x u du u u 0 w 0 1 r is obtained where is a bounded domain in rn with a lipschitz boundary max 1 p 1

the maximum principle of alexandrov for very weak solutions abstract we extend the classical maximal principle of alexandrov to very weak solutions of the elliptic equation i j 1 n a i j x 2 u x i x j f

a discrete linearizability test based on multiscale analysis in this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad graph by the multiple scale reduction around their harmonic solution we show that the a1 a2 and a3 linearizability conditions restrain the number of the parameters which enter into the equation a subclass of the equations which pass the a3 c integrability conditions can be linearized by a mobius transformation

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

lyapunov function and spectrum comparison for a reaction diffusion system with mass conservation abstract we are dealing with a two component system of reaction diffusion equations with mass conservation in a bounded domain with the neumann boundary conditions we prove the global boundedness of the solution in l norm for t 0 under a condition and then the existence of a lyapunov function moreover by studying the linearized eigenvalue problem of a nonconstant equilibrium solution we provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation as an application of the comparison result we obtain that any stable equilibrium solution must be monotone if the space dimension is one it is also shown that a modified system with a new parameter which covers the present model possesses a lyapunov function

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

t minima and application to the convergence of some integral functionals with infinite energy minima abstract we consider integral functionals j as in 1 we study the calderon zygmund theory for infinite energy minima u t minima of j and the stability of the t minima with respect to the convergence in 22 with v w 0 1 p

on very weak solutions to a class of double obstacle problems abstract we prove the uniqueness of very weak solutions by hodge decomposition and obtain the existence by an approximation argument moreover we prove that the solution quasiminimizes the r dirichlet integral in k r and finally the stability of very weak solutions is established for varying obstacle functions

stationary fix caginalp equation with non local term abstract we study the fix caginalp equation describing non isothermal solid liquid phase transition first we formulate a thermally closed stationary state as a nonlinear eigenvalue problem with a non local term then some results on multiple existence stability and bifurcation of the solution are proven

nonlinear schrodinger type equations from multiscale reduction of pdes i systematic derivation in this article we begin a systematic investigation via multiscale expansions of nonlinear evolution pdes partial differential equations in this first article we restrict consideration to a single autonomous but otherwise generic pde in 1 1 variables space time of first order in time whose linear part is dispersive and to solutions dominated by a single plane wave satisfying the linear part of the pde the expansion parameter is an assumedly small coefficient multiplying this plane wave the main indeed asymptotically exact effect of the weak nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics which is generally described in appropriately defined coarse grained time and space variables by evolution equations of nonlinear schrodinger type a systematic analysis of such equations is presented corresponding to various assumptions on the resonances occurring for the first few harmonics

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

c integrable nonlinear partial differential equations iii a technique based on a change of dependent variables used in a previous paper to generate c integrable nonlinear partial differential equations pdes i e nonlinear pdes linearizable by an appropriate change of variables in 1 1 dimensions one time and one space variables is extended to the case of more than one space dimension several examples of evolution c integrable pdes in 1 2 dimensions one time and two space variables are exhibited

multiscale expansion of the lattice potential kdv equation on functions of an infinite slow varyness order we present a discrete multiscale expansion of the lattice potential korteweg de vries lpkdv equation on functions of an infinite order of slow varyness to do so we introduce a formal expansion of the shift operator on many lattices holding at all orders the lowest secularity condition from the expansion of the lpkdv equation gives a nonlinear lattice equation depending on shifts of all orders of the form of the nonlinear schrodinger equation

on a reverse estimate for hodge decompositions of p laplacian type operators let x p 2 be a p laplacian type operator and consider the hodge decomposition x du d h div h 0 a standard elliptic theory asserts that d q p 1 c du p 1q for all q p 1 there has been considerable recent interest in the validity of the reverse estimate du p 1q c d q p 1 for q p 1 in the regularity study of certain geometrical mappings in this paper we give a relatively new proof of a well known theorem that this reverse estimate holds for all q sufficiently close to the natural power p and also prove that the estimate holds for all q p 1 for certain special weak solutions u

c integrable nonlinear partial differentiation equations i a technique to generate c integrable nonlinear partial differentiation equations pdes i e nonlinear pdes linearizable by an appropriate change of variables is reported and several examples of such pdes are exhibited

gradient regularity for elliptic equations in the heisenberg group abstract we give dimension free regularity conditions for a class of possibly degenerate sub elliptic equations in the heisenberg group exhibiting super quadratic growth in the horizontal gradient this solves an issue raised in j j manfredi g mingione regularity results for quasilinear elliptic equations in the heisenberg group math ann 339 2007 485 544 where only dimension dependent bounds for the growth exponent are given we also obtain explicit a priori local regularity estimates and cover the case of the horizontal p laplacean operator extending some regularity proven in a domokos j j manfredi c 1 regularity for p harmonic functions in the heisenberg group for p near 2 in contemp math vol 370 2005 pp 17 23 in turn using some recent techniques of caffarelli and peral l caffarelli i peral on w 1 p estimates for elliptic equations in divergence form comm pure appl math 51 1998 1 21 the a priori estimates found are shown to imply the suitable local calderon zygmund theory for the related class of non homogeneous possibly degenerate equations involving discontinuous coefficients these last results extend to the sub elliptic setting a few classical non linear euclidean results t iwaniec projections onto gradient fields and l p estimates for degenerated elliptic operators studia math 75 1983 293 312 e dibenedetto j j manfredi on the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems amer j math 115 1993 1107 1134 and to the non linear case estimates of the same nature that were available in the sub elliptic setting only for solutions to linear equations

smallness and cancellation in some elliptic systems with measure data abstract in a bounded open subset r n we study dirichlet problems with elliptic systems involving a finite radon measure on r n with values into r n defined by div a x u x d u x in u 0 on where a i x y 1 n j 1 n a i j x y j with 1 n the equation index we prove the existence of a distributional solution u r n obtained as the limit of approximations by assuming i that coefficients a i j are bounded caratheodory functions ii ellipticity of the diagonal coefficients a i j and iii smallness of the quadratic form associated to the off diagonal coefficients a i j i e verifying a r staircase support condition with r 0 such a smallness condition is satisfied for instance in each one of these cases a a i j a j i skew symmetry b a i j is small c a i j may be decomposed into two parts the first enjoying skew symmetry and the second being small in absolute value we give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works a brezis s type nonexistence result is also given for general smooth elliptic hyperbolic systems

quasilinear elliptic systems with measure data abstract we study the existence of solutions of quasilinear elliptic systems involving n equations and a measure on the right hand side with the form i 1 n x i 1 n j 1 n a i j x u x j u in u 0 on where 1 n is the equation index is an open bounded subset of r n u r n and is a finite randon measure on r n with values in r n existence of a solution is proved for two different sets of assumptions on a examples are provided that satisfy our conditions but do not satisfy conditions required on previous works on this matter

multiscale analysis of discrete nonlinear evolution equations the method of multiscale analysis is constructed for discrete systems of evolution equations for which the problem is that of the far behaviour of an input boundary datum discrete slow space variables are introduced in a general setting and the related finite differences are constructed the method is applied to a series of representative examples the toda lattice the nonlinear klein gordon chain the takeno system and a discrete version of the benjamin bona mahoney peregrini equation among the resulting limit models we find a discrete nonlinear schrodinger equation with reversed spacetime a three wave resonant interaction system and a discrete modified volterra model

stability for p laplace type equation in a borderline case abstract we study the dirichlet problem for a p laplacian type operator in the setting of the orlicz zygmund space q log r n q 1 and 0 more precisely our aim is to establish under which assumptions on 0 existence and uniqueness of the solution are assured and to prove continuity and stability of the associated nonlinear operator

on non linear elliptic equation with weight abstract in this paper we prove the existence and uniqueness theorem of the dirichlet problem for certain non linear elliptic operators with muckenhoupt weight

higher integrability of very weak solutions of systems of p x laplacean type abstract we prove higher integrability of very weak solutions of a class of nonlinear elliptic systems whose model is the p x laplacean system div d u p x 2 d u div f p x 2 f i e we show that for small enough a function which solves the system and has bounded p x 1 energy satisfies d u p x l loc 1

the kundu eckhaus equation and its discretizations in this paper we show that the complex burgers and the kundu eckhaus equations are related by a miura transformation we use this relation to discretize the kundu eckhaus equation

nonlinear potential theory of elliptic systems abstract we give a summary of recent results from nonlinear potential theory focusing on linear and nonlinear potential estimates of solutions to non homogeneous equations and systems we start with the cases of quasilinear possibly degenerate equations of p laplacian type and passing through fully nonlinear elliptic equations finally move to systems in this last case we describe recent results implying potential estimates for the p laplacian system finally we describe results bridging nonlinear potential theory and classical partial regularity theory for general elliptic systems the main outcomes are new e regularity criteria involving both excess functionals and nonlinear potentials

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

regularity of solutions of degenerate a harmonic equations we study the regularity of solutions to degenerate a harmonic equations under suitable integrability assumptions on the ellipticity and growth coefficients in particular we show a self improving property of the gradient of the solutions extending previous results by gehring and iwaniec sbordone valid in the uniformly elliptic setting

multiple scale perturbation beyond the nonlinear schroedinger equation i abstract we consider the effect of weak nonlinearity on the propagation of one dimensional strongly dispersive waves in the standard quasi monochromatic approximation it is well known that the first order amplitude modulation satisfies the nonlinear schroedinger equation in slow variables with the assumption that this amplitude modulation has no discrete spectrum component i e no soliton we explore the higher order effects and provide a well defined procedure to compute all coefficients of the perturbative expansion as relatively bounded functions no secularity effects of slow variables the main effect of this procedure is that the leading amplitude modulation satisfies at higher orders the nonlinear schroedinger hierarchy of evolution equations in slow variables key ingredient is the integrability of the nonlinear schroedinger equation and its commuting symmetries

a distance between orbits that controls commutator estimates and invertibility of operators abstract a distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits thus proving a version of shneiberg s theorem for orbital methods the same machinery is used to extend the celebrated rochberg weiss commutator theorem to the setting of orbital methods it is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element

nonlinear commutators for the fractional p laplacian and applications we prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p laplacian this implies that solutions to certain degenerate nonlocal equations are higher differentiable also weakly fractional p harmonic functions which a priori are less regular than variational solutions are in fact classical as an application we show that sequences of uniformly bounded frac n s harmonic maps converge strongly outside at most finitely many points

asymptotic behavior of equilibrium states of reaction diffusion systems with mass conservation abstract we deal with a stationary problem of a reaction diffusion system with a conservation law under the neumann boundary condition it is shown that the stationary problem turns to be the euler lagrange equation of an energy functional with a mass constraint when the domain is the finite interval 0 1 we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d the ratio of the diffusion coefficient of the system tends to zero in view of a logarithmic function in the leading term of the potential we get to a scaling parameter satisfying the relation e d log 2 the main result shows that a sequence of minimizers converges to a dirac mass multiplied by the total mass and that by a scaling with the asymptotic profile exhibits a parabola in the nonvanishing region we also prove the existence of an unstable monotone solution when the mass is small

a unified theory of commutator estimates for a class of interpolation methods abstract a general family of interpolation methods is introduced which includes as special cases the real and complex methods and also the so called or g1 and g2 methods defined by peetre and gustavsson peetre derivation operators and translation operators r are introduced for all methods of this family a theorem is proved about the boundedness of the commutators t and t r for operators t which are bounded on the spaces of the pair to which the interpolation method is applied this extends and unifies results previously known for derivation and translation operators in the contexts of the real and complex methods other results deal with higher order commutators and also include an equivalence theorem i e it is shown that as previously known only for real interpolation spaces all these interpolation spaces have two different equivalent definitions in the style of the j method and k method auxiliary results which may also be of independent interest include the equivalence of lions schechter complex interpolation spaces defined using an annulus with the same spaces defined in the usual way using a strip

linearization through symmetries for discrete equations we show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete cole hopf transformation if the equation is linearizable then the symmetry provides the linearizing transformation at the end we present a few examples of applications for equations defined on four lattice points

convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations abstract we study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t div a x u f x u 0 on a bounded domain subject to dirichlet boundary and to initial conditions the data are supposed to satisfy suitable regularity and growth conditions our approach to the convergence result and decay estimate is based on the ojasiewicz simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates the abstract results and their applications are discussed also in the framework of orlicz sobolev spaces

on coercive variational integrals abstract it is well known that sequential weak lower semicontinuity of a variational integral f u f u x d x on the sobolev space w 1 p r n under a p growth condition on the integrand f is equivalent to quasiconvexity in the sense of morrey we show that coercivity on dirichlet classes likewise is equivalent to a quasiconvexity condition we also discuss some examples and extend a sequential weak lower semicontinuity result to the case of signed integrands

discrete reductive perturbation technique we expand a partial difference equation p e on multiple lattices and obtain the p e which governs its far field behavior the perturbative reductive approach is here performed on well known nonlinear p es both integrable and nonintegrable we study the cases of the lattice modified korteweg de vries mkdv equation the hietarinta equation the lattice volterra kac van moerbeke equation and a nonintegrable lattice kdv equation such reductions allow us to obtain many new p es of the nonlinear schrodinger type

very weak solutions to elliptic equations with singular convection term abstract we study dirichlet problem for a nonlinear equation with a drift term despite the presence of the singular convection term we establish existence and uniqueness of a solution in spaces larger than the natural one

regularity in oscillatory nonlinear elliptic systems we give uniform bounds in morrey sobolev spaces h1 for weak solutions ue to oscillatory nonlinear elliptic systems the main novelty is in the precise qualitative description of the e uniform morrey exponent in terms of the ellipticity ratio of the elliptic systems this is achieved by showing that weak solutions of general nonlinear elliptic systems with lipschitz coefficients are of class w2 p locally or globally depending on assumptions for an exponent p 2 that is independent of dimensions and for which p 2 is inversely proportional to the ellipticity ratio

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians