nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

ultradiffusion on the fractal branching koch curves ultradiffusion is defined on a family of fractal branching koch curves by an exact renormalization decimation transformation the anomalous long time behaviour of the autocorrelation function is obtained for the particular hierarchy the authors find a non universal crossover c1 m 1 2m and a universal one c2 1 2 as the effective temperature c is increased where m is the maximum ramification number of the fractal they also find that for large m the dynamic behaviour of ultradiffusion on the fractal is independent of the effective temperature c

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

linear oblique derivative problems for the uniformly elliptic hamilton jacobi bellman equation on demontre la resolubilite et l unicite des problemes aux valeurs limites generaux et de dirichlet

global solutions of shock reflection by large angle wedges for potential flow when a plane shock hits a wedge head on it experiences a reflectiondiffraction process and then a self similar reflected shock moves outward as the original shock moves forward in time experimental computational and asymptotic analysis has shown that various patterns of shock reflection may occur including regular and mach reflection however most of the fundamental issues for shock reflection have not been understood including the global structure stability and transition of the different patterns of shock reflection therefore it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection on the other hand there has been no rigorous mathematical result on the global existence and structural stability of shock reflection including the case of potential flow which is widely used in aerodynamics such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic hyperbolic type free boundary problems and corner singularity where an elliptic degenerate curve meets a free boundary in this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large angle wedges for potential flow the techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties

oblique boundary value problems for augmented hessian equations ii abstract in this paper we continue our investigations into the global theory of oblique boundary value problems for augmented hessian equations we construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented hessian is only assumed regular this enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case finally we give some applications to existence theorems which embrace standard hessian equations as special cases

unique continuation for δ and the c fefferman phong class we show that the strong unique continuation property holds for the inequality u v u where the potential v x satisfies the c fefferman phong condition in a certain range of p values we also deal with the situation of u x vanishing at infinity these are all consequences of appropriate carleman inequalities

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

on existence of solutions for a class of fully nonlinear noncoercive problems 0 arises in the three dimensional water wave theory see m shinbrot lo and also in the engineering problem of hydraulic fracturing of oil wells see j r canon and g h meyer 2 7 mathematically the model problem 1 is of interest since it represents one of the simplest noncoercive elliptic problems i e the lopatinski shapiro condition fails see 4 for which there is no general theory the field of fully nonlinear elliptic equations has recently attracted a great deal of attention see e g a recent paper of g m lieberman and n s trudinger 6 and the references therein to other work of the same authors as well as that of l evans n krylov p l lions and others it appears that the case of second order boundary operators is not well studied particularly for noncoercive problems we present an existence result for l which extends the one in 3 where we had considered the equation

two dimensional nonlinear boundary value problems for elliptic equations on considere la regularite limite des solutions du probleme aux valeurs limites non lineaire f x u du d 2 u 0 dans g x u du 0 sur pour des domaines a 2 dimensions f est uniformement elliptique et g depend de du on demontre des estimations de continuite pour des derivees premieres et secondes de u sous des hypotheses faibles pour la regularite de f g et

ultradiffusion the relaxation of hierarchical systems the authors show that the dynamics of relaxation in hierarchical structures leads to an anomalous decay process which they term ultradiffusion using renormalisation group techniques they explicitly calculate the behaviour of the autocorrelation function in a one dimensional hierarchical system and show its universal power law decay as a function of temperature the authors also demonstrate analytically the emergence of a hierarchy of time scales this phenomenon is relevant to systems ranging from macromolecules to computing structures

the nonlinear oblique derivative problem for quasilinear elliptic equations on considere des problemes aux valeurs limites pour une equation elliptique quasilineaire d ordre 2 qu 0 dans ou est un domaine borne de r n n 2 et q est donne sur c 2 par qu x a ij x u x du x d ij u x a x u x du x

the neumann problem of special lagrangian equations with supercritical phase abstract in this paper we establish the global c 2 estimates of the neumann problem of special lagrangian equations with supercritical phase and the existence theorem by the method of continuity

unique continuation and classification of blow up profiles for elliptic systems with neumann boundary coupling and applications to higher order fractional equations abstract in this paper we develop a monotonicity formula for elliptic systems with neumann boundary coupling proving unique continuation and classification of blow up profiles as an application we obtain strong unique continuation for some fourth order equations and higher order fractional problems

a partial answer to a conjecture of b simon concerning unique continuation in his survey paper on schrijdinger semigroups s b simon formulated the following conjecture see s p 5193 let ve kfp the local kato class of potentials then the time independent schrddinger operator h d v has the unique continuation property ucp by this it is meant that given any connected open subset 52 c r the only solution u of hu 0 in 52 that can vanish in an open subset 52 c sz is u 0 a measurable function v on r is said to belong to the kato class k if

free boundary and mixed boundary value problems for quasilinear elliptic equations tangential oblique derivative and degenerate dirichlet boundary problems abstract we present holder estimates and holder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector does not vanish at any point on the boundary we also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

existence of capillary gravity water waves with piecewise constant vorticity abstract in this paper we construct small amplitude periodic capillary gravity water waves with a piecewise constant vorticity distribution they describe water waves traveling on superposed linearly sheared currents that have different vorticities this is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump the solutions of the diffraction problem found by using local bifurcation theory are the desired solutions of the hydrodynamical problem

the conormal derivative problem for elliptic equations of variational type on considere des solutions du probleme qu 0 dans nu 0 sur ou est un domaine borne dans r n et q et n sont definis par qu x div a x u x du x b x u x du x x nu x a x u x du x x x x

quasi linear variable exponent boundary value problems with wentzell robin and wentzell boundary conditions let p c0 1 be such that 1 p p 1 p p let rn rn be a bounded w1 p w1 p extension domain and let be an upper d ahlfors measure supported on with d n p n d n p n we investigate the solvability of a class of quasi linear boundary value problems involving the p p laplace and p p laplace beltrami operators and either classical wentzell robin boundary conditions p ud u p 2 u dhn 1 u p 2ud 0on or general fully wentzell type boundary conditions of the form p ud p ud u p 2 u dhn 1 u p 2ud 0on where l d l d is such that infx x 0 for some constant 0 0 0 0 and u dhn 1 denotes the generalized p p normal derivative on in the interpretative sense we prove that the realization of the p p laplace operator with both of the above boundary conditions generate nonlinear ultracontractive submarkovian c0c0 semigroups on l2 dx l2 d l2 dx l2 d and hence their associated first order cauchy problems are both well posed on lq dx lq d lq dx lq d for all measurable function q with 1 q q 1 q q in addition we investigate the associated quasi linear elliptic problem with general wentzell boundary conditions and obtain existence uniqueness and global regularity of weak solutions to this equation

the viscosity method for the homogenization of soft inclusions in this paper we consider periodic soft inclusions te with periodicity e where the solution ue satisfies semi linear elliptic equations of non divergence in omega epsilon omega setminus overline t epsilon with neumann data on partial t mathfrak a the difficulty lies in the non divergence structure of the operator where the standard energy method which is based on the divergence theorem cannot be applied the main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of ue referred to as u as e approaches to zero we introduce the concept of a compatibility condition between the equation and the neumann condition on the boundary for the existence of uniformly bounded periodic first correctors the concept of a second corrector is then developed to show that the limit u is the viscosity solution of a homogenized equation

regularity for fully nonlinear elliptic equations with oblique boundary conditions in this paper we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions in particular we derive the pointwise c c 1 and c 2 regularity as byproducts we also prove the a b p maximum principle harnack inequality uniqueness and solvability of the equations

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

almgren type monotonicity methods for the classification of behaviour at corners of solutions to semilinear elliptic equations a monotonicity approach to the study of the asymptotic behavior near corners of solutions to semilinear elliptic equations in domains with a conical boundary point is discussed the presence of logarithms in the first term of the asymptotic expansion is excluded for boundary profiles sufficiently close to straight conical surfaces

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

ultradiffusion on multifurcating hierarchical structures diffusion on a trifurcating hierarchical structure is studied through an exact renormalisation group procedure the non universal time scaling exponent of the autocorrelation function is obtained and compared with that of the simplest 1d ultradiffusion model when a biasing term is introduced it is found that the transition from the power law to exponential decay of the autocorrelation function is independent of the furcating number of the hierarchy a general z furcating hierarchical system is also discussed briefly

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

linear second order elliptic equations with venttsel boundary conditions we prove the existence of solutions of quasilinear second order elliptic equations with quasilinear venttsel boundary conditions in both classical and weak sense under natural structure conditions

solvability of quasilinear elliptic equations with nonlinear boundary conditions ii abstract it is shown that solvability of the second order quasilinear elliptic equation qu 0 in with first order nonlinear boundary condition nu 0 on can be inferred from appropriate estimates on solutions of the problem qu in nu 0 on as varies over a suitable function class this result improves previous work of the author where estimates were required on solutions of qu in nu on as varies over some function space the value of this improvement is demonstrated by some examples

weighted doubling properties and unique continuation theorems for the degenerate schrodinger equations with singular potentials abstract let u be the weak solution to the degenerate schrodinger equation with singular coefficients in lipschitz domain as following div w x a x u x v x u x w x 0 where a x is a real symmetric matrix function satisfying the elliptic condition and the lipschitz continuity w x is an a 2 weight function of muckenhoupt class and v x is the fefferman phong s potential the weighted doubling properties and unique continuations for the weak solution u in the interior of any domains as well as at the boundary of some lipschitz domains are derived in this paper

three dimensional steady subsonic euler flows in bounded nozzles abstract the existence and uniqueness of three dimensional steady subsonic euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit if in addition the normal component of the voriticity and the variation of bernoulli s function at the entrance are both zero then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number as the magnitude of the normal component of the momentum approaches the critical number the associated flows converge to a subsonic sonic flow furthermore when the normal component of vorticity and the variation of bernoulli function are both small the existence and uniqueness of subsonic euler flows with non zero vorticity are established the proof of these results is based on a new formulation for the euler system a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions detailed study for a linear div curl system and delicate estimate for the transport equations

study of the 1d biased and 2d ultradiffusion models the authors study within a renormalisation group scheme two possible generalisations of a one dimensional 1d diffusion model with a hierarchy of energy barriers recently proposed they consider the same 1d model plus a biasing term and a 2d version of it in the first case they find for a suitable value of the biasing force a new transition in the long time behaviour of the autocorrelation function for the 2d model through an approximate treatment they obtain the corresponding anomalous exponent which characterises its relational behaviour

on application of the monotone iteration scheme to noncoercive elliptic and hyperbolic problems on montre que le schema d iteration monotone peut s appliquer avec succes aux equations autres que coercives elliptiques et hyperboliques

gradient bounds for solutions of nonuniformly elliptic oblique derivative problems introduction in the theory of quasilinear elliptic equations there are three archetypical forms which seem to embody the various possible behaviors of solutions laplace s equation or more generally a uniformly elliptic equation the minimal surface equation and a sort of false minimal surface equation sii diudiu d u 0 for each of these equations the behavior of solutions to the dirichlet problem is wellunderstood see for example 3 5 10 11 additionally uniformly elliptic conormal problems were studied in 5 chapter lo and minimal surface type conormal problems xvere studied in l 13 61 the class of equations considered in 6 was much broader than those in l 131 but did not include the false minimal surface equation finally uniformly elliptic oblique derivative problems were studied in 7 91 while minimal surface type oblique derivative problems are the subject of 4 here we examine the oblique derivative problem on a smooth domain q c 2

stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations abstract we develop a method that works in general product riemannian manifold to decompose the three dimensional steady full compressible euler system which is of elliptic hyperbolic composite mixed type for subsonic flows the method is applied to show stability of spherically symmetric subsonic flows and transonic shocks in space r 3 under multidimensional perturbations of boundary conditions

global regularity of solutions of nonlinear second order elliptic and parabolic differential equations on considere des solutions du probleme de dirichlet pour une equation elliptique totalement non lineaire dans un domaine borne lisse et du probleme parabolique analogue

a global subsonic solution to an interacting transonic shock for the self similar nonlinear wave equation abstract we present the existence of the subsonic solution to a two dimensional riemann problem governed by a self similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle by utilizing the barrier methods and iterative methods we show the well posedness of the transonic shock in the entire subsonic region and thus establish the global solution this result does not rely on any smallness of riemann data

weingarten hypersurfaces with prescribed gradient image we prove the existence of globally smooth convex solutions u of a class of curvature equations subject to the boundary condition du omega omega where omega and omega are smooth uniformly convex domains in mathbb r n the results generalize some of our previous work on the two dimensional case and on hessian equations in all dimensions

regularity of solutions of nonlinear elliptic boundary value problems on utilise des estimations de type schauder pour ameliorer la regularite supposee des solutions de problemes aux limites non lineaires elliptiques

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

the neumann problem for equations of monge ampere type we prove the existence of a classical solution to a neumann type problem for nonlinear elliptic equations of monge ampere type the methods depend upon the establishment of a priori derivative estimates up to order two and yield a sharp result for the equation of prescribed gauss curvature we also present applications to asymptotic problems which approach the non unique case and maximum principles for linear equations of aleksandrov bakelman type

c1 domains and unique continuation at the boundary it is shown that the square of a nonconstant harmonic function u that either vanishes continuously on an open subset v contained in the boundary of a dini domain or whose normal derivative vanishes on an open subset v in the boundary of a c1 1 domain in d satisfies the doubling property with respect to balls centered at points q v under any of the above conditions the module of the gradient of u is a b2 d weight when restricted to v and the hausdorff dimension of the set of points q v u q 0 is less than or equal to d 2 these results are generalized to solutions to elliptic operators with lipschitz second order coefficients and bounded coefficients in the lower order terms 1997 john wiley sons inc