We find nontrivial smooth solutions for nonlinear elliptic Dirichlet problems driven by the p-Laplacian (1 < p < ∞), when one sided resonance occurs at the principal spectral interval.

A class of nonlinear elliptic and parabolic optimal control problems with mixed control-state constraints is considered. Extending a method known for the control of ordinary differential equations to the case of PDEs, the Yosida-Hewitt theorem is applied to show that the Lagrange multipliers are functions of certain Lp-spaces. By bootstrapping arguments, under natural assumptions, optimal contr...

We present an implementation of Böhmer’s finite element method for fully nonlinear elliptic partial differential equations on convex polygonal domains, based on a modified Argyris element and BernsteinBézier techniques. Our numerical experiments for several test problems, involving the classical Monge-Ampère equation and an unconditionally elliptic equation, confirm the convergence and error bo...

The asymptotic behavior of the solution of a class of elliptic problems modeling diffusion in some periodic perforated media is analyzed. We consider, at the microscale, an elliptic equation with various nonlinear conditions prescribed on the boundary of the perforations and we prove that the effective behavior of the solution of such a problem is governed by another elliptic equation, which, d...

Contents 1. Introduction. 2. Models of PDE 's in the 18th and 19th century. 3. Methods of calculating solutions in the 19th century. 4. Developments of rigorous theories of solvability in the last decades of the 19th century. 5. The period 18901900: the beginning of modern PDE and the work of Poincare. 6. The Hilbert programs. 7. S. Bernstein and the beginning of a priori estimates. 8. Solvabil...

CONTENTS Introduction. 0. Pseudodifferential operators and linear PDE. §0.1 The Fourier integral representation and symbol classes §0.2 Schwartz kernels of pseudodifferential operators §0.3 Adjoints and products §0.4 Elliptic operators and parametrices §0.5 L 2 estimates §0.6 Gårding's inequality §0.7 The sharp Gårding inequality §0.8 Hyperbolic evolution equations §0.9 Egorov's theorem §0.10 M...

‎This paper is devoted to get a ground state solution for a class of nonlinear elliptic equations with fast increasing weight‎. ‎We apply the variational methods to prove the existence of ground state solution‎.