We study the problem of minimizing ${\int }_{\Omega }F\left(Du\left(x\right)\right)dx$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi$ on $\Gamma :=\partial \Omega$. The lagrangian $F$ and the domain $\Omega$ are assumed convex. A new type of hypothesis on the boundary function $\phi$ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space  [0, +∞) × ℝn and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.

In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...

We discuss the existence of solutions for a certain generalization of the membrane equation and their continuous dependence on function parameters. We apply variational methods and consider the PDE as the Euler-Lagrange equation for a certain integral functional, which is not necessarily convex and coercive. As a consequence of the duality theory we obtain variational principles for our problem and some numerical results concerning approximation of solutions.

A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given.

A necessary and sufficient condition for the continuous extendibility of a solution of the third problem for the Laplace equation is given.

We study the semilinear problem with the boundary reaction $$-\Delta u+u=0\text{in}\Omega ,\frac{\partial u}{\partial \nu }=\lambda f\left(u\right)\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$, $N\ge 2$, is a smooth bounded domain, $f:[0,\infty )\to (0,\infty )$ is a smooth, strictly positive, convex, increasing function which is superlinear at $\infty$, and $\lambda >0$ is a parameter. It is known that there exists an extremal parameter ${\lambda }^{*}>0$ such that a classical minimal solution exists for $\lambda <{\lambda }^{*}$, and there is no solution for $\lambda >{\lambda }^{*}$. Moreover, there is a unique weak solution $u^{*}$ corresponding to the parameter $\lambda ={\lambda }^{*}$. In this paper, we continue to study the spectral properties of $u^{*}$ and show...

We give some counterexamples concerning the regularity of the first (resp. second) derivatives of solutions of linear second order elliptic partial differential equations in divergence form (resp. in non-divergence form).

For the Schrödinger equation, $(i{\partial }_t+\nabla )u=0$ on a torus, an arbitrary non-empty open set $\Omega$ provides control and observability of the solution: $∥u{\left|{}_{t=0}\right.∥}_{L^2\left({𝕋}^2\right)}\le K_T{∥u∥}_{L^2([0,T]\times \Omega )}$. We show that the same result remains true for $(i{\partial }_t+\nabla -V)u=0$ where $V\in L^2\left({𝕋}^2\right)$, and ${𝕋}^2$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V\in C\left({𝕋}^2\right)$ and conjectured for $V\in L^{\infty }\left({𝕋}^2\right)$. The higher dimensional generalization remains open for $V\in L^{\infty }\left({𝕋}^n\right)$.

The paper is devoted to the convergence analysis of a well-known cell-centered Finite Volume Method (FVM) for a convection-diffusion problem in ${ℝ}^2$. This FVM is based on Voronoi boxes and exponential fitting. To prove the convergence of the FVM, we use a new nonconforming Petrov-Galerkin Finite Element Method (FEM) for which the system of linear equations coincides completely with that of the FVM. Thus, by proving convergence properties of the FEM we obtain similar ones for the FVM. For the error...