Regularity for a more general class of quasilinear elliptic equations

On etudie la regularite limite des solutions faibles de l'equation divA(x,u,Du)+B(x,u,Du)=0 avec des conditions aux limites conormales et de Dirichlet

Boundary regularity for solutions of degenerate elliptic equations

In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ω ⊂ ℝ
 n
 ,n ⩾ 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of − Δu + β(u) = f withβ a nondecreasing function ℝ → ℝ,β(0)=0, andf⩾0 a.e. in Ω if and only if the integral∫(β(s)s)
−1/2
ds diverges ats=0+. We extend the result to more general equations, in particular to − Δ
 p
 
u + β(u) =f where Δ
 p
 (u) = div(|Du|

p-2
Du), 1 <p < ∞. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.

A Strong Maximum Principle for some quasilinear elliptic equations

(1991). The natural generalizationj of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations. Communications in Partial Differential Equations: Vol. 16, No. 2-3, pp. 311-361.

The natural generalizationj of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations

(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

(1983). On The Dirichletproblem for Quasilinear Equations. Communications in Partial Differential Equations: Vol. 8, No. 7, pp. 773-817.

On The Dirichletproblem for Quasilinear Equations

THE BEHAVIOUR near singular boundary points of elliptic boundary value problems is of particular interest for physical applications (cf. [l]) an numerical computations (cf. [2]). For d linear equations this subject is almost completely treated by Grisvard [3] and Kondratjev [4]. They proved that the solution can be split near a plane corner or conical point into a smooth part and a linear combination of certain singular functions. For the model problem

On quasilinear boundary value problems in domains with corners

On considere la notion de minimum d'integrales variationnelles en liaison avec les solutions d'equations elliptiques et de systemes sous forme divergence

Peter Tolksdorf

Papers

Regularity for a more general class of quasilinear elliptic equations

On The Dirichletproblem for Quasilinear Equations

On quasilinear boundary value problems in domains with corners

Remarks on quasi(sub)minima

A New Proof of a Regulatory Theorem.