a priori bounds for semilinear equations and a new class of critical exponents for lipschitz domains abstract a priori bounds for positive very weak solutions of semilinear elliptic boundary value problems u f x u on a bounded domain r n with u 0 on are studied where the nonlinearity 0 f x s grows at most like s p if is a lipschitz domain we exhibit two exponents p and p which depend on the boundary behavior of the green function and on the smallest interior opening angle of we prove that for 1 p p all positive very weak solutions are a priori bounded in l for p p we construct a nonlinearity f x s a x s p together with a positive very weak solution which does not belong to l finally we exhibit a class of domains for which p p for such domains we have found a true critical exponent for very weak solutions in the case of smooth domains p p n 1 n 1 is an exponent which is well known from classical work of brezis turner h brezis r e l turner on a class of superlinear elliptic problems comm partial differential equations 2 1977 601 614 and from recent work of quittner souplet p quittner ph souplet a priori estimates and existence for elliptic systems via bootstrap in weighted lebesgue spaces arch ration mech anal 174 2004 49 81

a theoretical and computational framework for anisotropic continuum damage mechanics at large strains abstract the main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains based on the concept of a fictitious isotropic undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient then the corresponding finger tensor denoted as damage metric constructs a second order internal variable due to the principle of strain energy equivalence with respect to the fictitious effective space and the standard reference configuration the free energy function can be computed via push forward operations within the nominal setting referring to the framework of standard dissipative materials associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation the numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account finally some numerical examples demonstrate the applicability of the proposed framework

universal bounds for global solutions of a diffusion equation with a nonlocal reaction term we consider the nonlinear heat equation with nonlocal reaction term in space ut u up in smoothly bounded domains we prove the existence of a universal bound for all nonnegative global solutions of this equation moreover in contrast with similar recent results for equations with local reaction terms this is shown to hold for all p 1 as an interesting by product of our proof we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction

optimal conditions for l regularity and a priori estimates for semilinear elliptic systems abstract in this paper we present a bootstrap procedure for semilinear elliptic systems with n 3 components combining with the l p l q estimates it yields the optimal l regularity conditions for the three well known types of weak solutions h 0 1 solutions l 1 solutions and l 1 solutions thanks to the linear theory in l p it also yields the optimal conditions for a priori estimates for l 1 solutions based on the a priori estimates we improve known existence theorems for some classes of elliptic systems

semilinear reaction diffusion systems of several components abstract the homogeneous dirichlet boundary value problem u it u i j 1 n u j p ij i 1 2 n in a bounded domain r n is considered where p ij 0 1 i j n are constants denote by i the identity matrix and p pij which is assumed to be irreducible we find out that whether or not i p is a so called m matrix plays a fundamental role in the blow up theorems

on the spatial formulation of anisotropic multiplicative elasto plasticity abstract this contribution aims at a thermodynamically consistent and modular formulation of anisotropic multiplicative elasto plasticity embedded in the well established framework of non standard dissipative materials we adopt as a key ingredient the introduction of additional symmetric arguments typically structural tensors into the relevant scalar valued isotropic tensor functions on this basis the fundamental covariance relation leads us to a formulation which allows itself to be set up directly in terms of spatial arguments and thus guarantees a convenient implementation within a finite element setting

optimal conditions for a priori estimates for semilinear elliptic systems with two components abstract in this paper we present a new bootstrap procedure for elliptic systems with two unknown functions when combined with the l p l q estimates it yields the optimal l regularity conditions for the three well known types of weak solutions h 0 1 solutions l 1 solutions and l 1 solutions thanks to the linear theory in l p it also yields the optimal conditions for a priori estimates for l 1 solutions based on the a priori estimates we improve known existence theorems for some classes of elliptic systems

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

universal bounds for global solutions of a forced porous medium equation we show that in a smooth bounded domain rn n 2 all global nonnegative solutions of ut um up with zero boundary data are uniformly bounded in by a constant depending on p and but not on u0 provided that 1 m p n 1 n 1 m furthermore we prove an a priori bound in l 0 depending on u0 l under the optimal condition 1 m p n 2 n 2 m

optimal regularity conditions for elliptic problems via l p_ delta spaces we consider positive solutions of the dirichlet problem for nonlinear elliptic equations where the nonlinearity is assumed to satisfy the polynomial upper growth condition f x u c 1 up it is known from a classical work of brezis and turner that the condition p pbt for a suitable nonlinearity f x u a x up with a x 0 and a l we prove the existence of an unbounded weak solution we next consider the case of systems and show that the polynomial growth conditions for a priori estimates recently obtained by quittner and the author are also optimal our results are strongly connected with the regularity theory of the laplace operator in the spaces l p delta omega the lebesgue spaces weighted by the distance to the boundary as a by product we in turn establish the optimality of the known linear l p delta regularity estimates our proofs are based on the construction of a solution of the laplace equation with a suitable boundary singularity with conical support and we use recent results on the boundary behavior of heat kernels

linear and nonlinear heat equations in l q_ delta spaces and universal bounds for global solutions we develop a theory of both linear and nonlinear heat equations in the weighted lebesgue spaces l q delta where delta is the distance to the boundary in particular we prove an optimal l q delta l r delta estimate for the heat semigroup and we establish sharp results on local existence uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces this theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems namely under certain assumptions we prove that any global solution is uniformly bounded for t geq tau 0 by a universal constant independent of the initial data in all previous results the bounds for global solutions were depending on the initial data

numerical differentiation for non trivial consistent tangent matrices an application to the mrs lade model in reference 1 the authors have shown that numerical differentiation is a competitive alternative to analytical derivatives for the computation of consistent tangent matrices relatively simple models were treated in that reference the approach is extended here to a complex model the mrs lade model 2 3 this plastic model has a cone cap yield surface and exhibits strong coupling between the flow vector and the hardening moduli because of this derivating these quantities with respect to stresses and internal variables the crucial step in obtaining consistent tangent matrices is rather involved numerical differentiation is used here to approximate thesederivatives theapproximatedderivativesarethenused1 tocomputeconsistenttangentmatrices globalproblem and2 tointegratetheconstitutiveequation at each gauss point local problem with the newton raphson method the choice of the stepsize i e the perturbation in the approximation schemes based on the concept of relative stepsize poses no difficulties in contrast to previous approaches for the mrs lade model quadratic convergence is achieved for both the local and the global problems the computational efficiency cpu time and robustness of the proposed approach is illustrated by means of several numerical examples where the major relevant topics are discussed in detail

on the boundedness of global solutions of abstract semilinear parabolic equations abstract in this paper we consider the question of the long time behavior of solutions of the initial value problem for evolution equations of the formdu dt au t f u t in a banach space whereais the infinitesimal generator of an analytic semigroup andfis a nonlinear function such that the initial value problem possesses nonglobal solutions for some initial data we give sufficient conditions to ensure that global solutions on 0 are necessarily bounded in a suitable norm we give several examples of problems to which the abstract result applies

weak eigenfunctions for the linearization of extremal elliptic problems abstract we consider the semilinear elliptic problem formula where is a nonnegative parameter and g is a positive nondecreasing convex nonlinearity there exists a value of the parameter which is extremal in terms of existence of solution we study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter in some cases this linearized problem has discrete and positive h 1 0 spectrum however we prove that there always exists a positive weak eigenfunction in l 1 with eigenvalue zero for this linearized problem the zero l 1 eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for finally we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when is the unit ball and g u e u or g u 1 u p

boundedness of global solutions of nonlinear diffusion equations it is well known that there exist choices of u0 for which the corresponding solutions tend to zero as t cc and other choices for which the solutions blow up in finite time if we are interested in other types of behavior we may consider the possibility p u t u exists globally

the exact derivative of the exponential of an unsymmetric tensor an exact series representation for the derivative of the exponential of a generic unsymmetric tensor is derived the computer implementation of the derived formula is straightforward and allows the computation of the derivative of the tensor exponential to any desired degree of accuracy in practice the accuracy of the computed derivatives will be limited by machine precision the application of the proposed formula to the numerical treatment of anisotropic rate dependent and rate independent finite single crystal plasticity is outlined

numerical computation of algorithmic consistent tangent moduli in large strain computational inelasticity abstract an algorithm for the numerical computation of so called algorithmic or consistent tangent moduli in finite inelasticity is presented these moduli determine the sensitivity of algorithmic expressions for stresses with respect to the change in total deformation they serve as iteration operators by application of newton type solvers for the iterative solution of non linear initial boundary value problems in finite inelasticity the underlying concept of the numerical computation is a perturbation technique based on a forward difference approximation which reduces the computation of the tangent moduli to a multiple stress computation the algorithmic procedure is material independent and surprisingly simple it is outlined for a lagrangian as well as an eulerian geometric setting and applied to model problems of finite elasticity and finite elastoplasticity

a priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient abstract we study nonnegative solutions of the boundary value problem p u c x u x u 2 h x u h 0 1 l where is a smooth bounded domain of r n c l h l r for some r n 2 and c h 0 our main motivation is to study the semidefinite case namely unlike in previous work on the subject we do not assume to be uniformly positive in nor even positive everywhere in space dimensions up to n 5 we establish uniform a priori estimates for weak solutions of p when 0 is bounded away from 0 this is proved under the assumption that the supports of and c intersect a condition that we show to be actually necessary and in some cases we further assume that is uniformly positive on the support of c and or some other conditions as a consequence of our a priori estimates assuming that p 0 has a solution we deduce the existence of a continuum c of solutions such that the projection of c onto the axis is an interval of the form 0 a for some a 0 and that the continuum c bifurcates from infinity to the right of the axis 0 in particular for each 0 small enough problem p has at least two distinct solutions

initial blow up of solutions of semilinear parabolic inequalities abstract we study classical nonnegative solutions u x t of the semilinear parabolic inequalities 0 u t u u p in 0 1 where p is a positive constant and is a bounded domain in r n n 1 we show that a necessary and sufficient condition on p for such solutions u to satisfy a pointwise a priori bound on compact subsets k of as t 0 is p 1 2 n and in this case the bound on u is max x k u x t o t n 2 as t 0 if in addition is smooth u satisfies the boundary condition u 0 on 0 1 and p 1 2 n then we obtain a bound for u on the entire set as t 0

estimates for green function and poisson kernels of higher order dirichlet boundary value problems pointwise estimates are derived for the kernels associated to the polyharmonic dirichlet problem on bounded smooth domains as a consequence one obtains optimal weighted l p l q regularity estimates for weights involving the distance function

attractors for non autonomous parabolic problems with singular initial data in this paper we study the asymptotic behavior of solutions of non autonomous parabolic problems with singular initial data we first establish the well posedness of the equation when the initial data belongs to lr 1 r and w1 r 1 r n respectively when the initial data belongs to lr we establish the existence of uniform attractors in lr for the family of processes with external forces being translation bounded but not translation compact in llocp r lr when we consider the existence of uniform attractors in h01 the solution of equation lacks the higher regularity so we introduce a new type of solution and prove the existence result for the long time behavior of solutions of the equation in w1 r we only obtain the uniform attracting property in the weak topology

a priori estimates and existence for elliptic systems via bootstrap in weighted lebesgue spaces we present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains this method is based on a bootstrap procedure used alternatively on each component in the scale of weighted lebesgue spaces lp lp x dx where x is the distance to the boundary using this method we significantly improve the known existence results for various classes of elliptic systems

optimal control problems with final observation governed by explosive parabolic equations we study optimal control problems with final observation the governing parabolic equations or systems involve superlinear nonlinearities and their solutions may blow up in finite time our proof of the existence regularity and optimality conditions for an optimal pair is based on uniform a priori estimates for the approximating solutions our conditions on the growth of the nonlinearity are essentially optimal in particular we also solve a long standing open problem of lions concerning singular systems

initial blow up rates and universal bounds for nonlinear heat equations we establish a universal upper bound on the initial blow up rate for all positive classical solutions of the dirichlet problem for the nonlinear heat equation ut u upon 0 t where p 1 and is a smoothly bounded domain in rn namely we show that u t c p t t on 0 t 2 for some n p 0 this is proved for all subcritical p i e p n 2 n 2 if n 4 and under a stronger assumption on p if n 5 as a consequence we improve the known results on universal bounds for global solutions furthermore if p n 3 n 1 then we may take n 1 2 and we show that this value of is optimal interestingly the rate can be faster than the previously known maximal initial blow up rate of the cauchy problem applications to universal blow up estimates at t t are given the neumann problem is also considered and we obtain the estimate on the initial rate for all subcritical p up to dimension n 6

a consistent time fe method for large strain elasto plasto dynamics in the present paper energy consistent momentum conserving time stepping schemes for geometrically nonlinear multiplicative elasto plasto dynamics referred to as ecmc cg methods are developed in this context the entire discretisation procedure of the global equations of motion is based on galerkin methods in space and time whereby special emphasis is placed on the desired conservation properties related to the approximation of time integrals to guarantee energy consistency also for higher order finite elements in time a thermodynamically motivated enhanced stress tensor which defines a non standard quadrature rule is derived for elasto plastic material behaviour concerning the integration of the local evolution equations in time exemplarily a well established first order accurate exponential update is applied and adapted according to the global time stepping scheme based on linear finite elements in time the performance of the resulting scheme is demonstrated by means of several representative numerical examples whereby it is shown that global energy consistency is guaranteed exactly within the calculation accuracy by applying the proposed concepts

geometrically non linear anisotropic inelasticity based on fictitious configurations application to the coupling of continuum damage and multiplicative elasto plasticity the objective of this contribution is the formulation and algorithmic treatment of a phenomenological framework to capture anisotropic geometrically non linear inelasticity in addition to the intermediate configuration of multiplicative elasto plasticity we further introduce two microscopic configurations of lagrangian and eulerian type which characterize the so called fictitious undamaged material this kinematical framework enables us to apply two well established postulates based on standard terminology in non linear continuum mechanics concerning the free energy function the postulate of strain energy equivalence is adopted and in view of the plastic dissipation potential the concept of effective stress is a natural outcome of the underlying kinematical assumptions finally we focus on the integration technique for the class of obtained evolution equations and present numerical examples for a prototype model to underline the applicability of the proposed framework copyright 2003 john wiley sons ltd

numerical differentiation for local and global tangent operators in computational plasticity in this paper numerical differentiation is applied to integrate plastic constitutive laws and to compute the corresponding consistent tangent operators the derivatives of the constitutive equations are approximated by means of difference schemes these derivatives are needed to achieve quadratic convergence in the integration at gauss point level and in the solution of the boundary value problem numerical differentiation is shown to be a simple robust and competitive alternative to analytical derivatives quadratic convergence is maintained provided that adequate schemes and stepsizes are chosen this point is illustrated by means of some numerical examples