Sharp a priori estimates for a class of nonlinear elliptic equations with lower order terms

Annali di Matematica Pura ed Applicata (1923 -) - We prove sharp a priori estimates for second-order quasi-linear elliptic operators in divergence form with a first-order term. Such estimates are...     

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.     

Higher order nonlinear elliptic equations

Methods of solution of boundary problems for divergent and nondivergent higher order nonlinear elliptic equations are described. Applications of topological methods to the study of general nonlinear boundary problems are given. Results on a priori estimates and properties of generalized solutions are cited.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 37, pp. 3–87, 1990.     

Dirichlet problem for elliptic equations with nonlinear first order terms: a comparison result

Summary We deal with the Dirichlet problem for elliptic equations with a nonlinearity involving the gradient of the solution. By symmetrization techniques, we reduce the problem of finding sharp estimates of solutions to an analogous problem for ordinary differential equations.Lavoro svolto nell'ambito del G.N.A.F.A. con parziale contributo del M.P.I.     

Upper and lower bounds for higher order eigenvalues of some semilinear elliptic equations

We prove upper and lower bounds on the eigenvalues (as the norm of the eigenfunction tends to zero) in bifurcation problems for a class of semilinear elliptic equations in bounded domains of R^N. It is shown that these bounds are computable in terms of the eigenvalues of the associated linear equation.     

On a class of non-uniformly elliptic equations

In this paper we study the existence and uniqueness of both weak solutions and entropy solutions for the Dirichlet boundary value problem of a class of non-uniformly elliptic equations. A comparison result is also discussed. Some well-known elliptic equations are the special cases of this equation.     

Some elliptic problems with singular natural growth lower order terms

In this paper we deal with a nonlinear elliptic problem, whose model is

     

An overdetermined problem in nonlinear elliptic equations of second order

We define some functionals involvingu(x) andx'u _i , whereu(x) is a classical solution of the equation (q ^p-2 u _i ) _i +k(u)q ^p =0,p > 1, and prove that such functionals satisfy a second order elliptic differential equation. By a suitable choice of such functionals we investigate an overdetermined problem.Partially supported by Regione Autonoma della Sardegna.     

Removable singularities of some strongly nonlinear elliptic equations

Let Ω be some open subset of ?^N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? satisfying some power like growth assumption, then any u∈L _loc ^∞ (Ω′) satisfying $$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$ , remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ?⁺ and above on ?^?, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L _loc ^∞ (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ .     

Global gradient estimates for non-uniformly elliptic equations

We consider a nonlinear and non-uniformly elliptic problem in divergence form on a bounded domain. The problem under consideration is characterized by the fact that its ellipticity rate and growth radically change with the position, which provides a model for describing a feature of strongly anisotropic materials. We establish the global Calderón–Zygmund type estimates for the distributional solution in the case that the boundary of the domain is of class \(C^{1,\beta }\) for some \(\beta >0\).     

Existence results for quasilinear elliptic equations with multivalued nonlinear terms

In this paper we study the existence of solutions u ∈ \({{W}^{1,p}_{0}}\) (Ω) with △ _p u ∈ L ²(Ω) for the Dirichlet problem 1 $$ \left\{ \begin{array} [c]{l}-\triangle_{p}u\left( x\right) \in-\partial{\Phi}\left( u\left( x\right) \right) +G\left( x,u\left( x\right) \right) ,x\in{\Omega},\\ u\mid_{\partial{\Omega}}=0, \end{array} \right. $$ where Ω ? R ^N is a bounded open set with boundary ?Ω, △ _p stands for the p?Laplace differential operator, ?Φ denotes the subdifferential (in the sense of convex analysis) of a proper convex and lower semicontinuous function Φ and G : Ω × R → 2^R is a multivalued map. We prove two existence results: the first one deals with the case where the multivalued map u ? G(x, u) is upper semicontinuous with closed convex values and the second one deals with the case when u ? G(x, u) is lower semicontinuous with closed (not necessarily convex) values.