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### A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### A differential inclusion : the case of an isotropic set

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^{2}$ that satisfies$\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \phantom{\rule{85.35826pt}{0ex}}\hfill \end{array}\right\$where $\Omega$ is an open set of ${ℝ}^{2}$ and $E$ is a compact isotropic set of ${ℝ}^{2×2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

### A differential inclusion: the case of an isotropic set

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^{2}$ that satisfies $\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right\$ where Ω is an open set of ${ℝ}^{2}$ and E is a compact isotropic set of ${ℝ}^{2×2}$. We will show an existence theorem under suitable hypotheses on φ.

### A first order partial differential equation with an integral boundary condition

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro si considera un’equazione alle derivate parziali del primo ordine con una condizione sulla frontiera di tipo integrale. Si studia resistenza, l'unicità e il comportamento asintotico delle soluzioni.

### A general existence theorem for differential inclusions in the vector valued case.

Portugaliae Mathematica. Nova Série

### A new approach to the existence of almost everywhere solutions of nonlinear PDEs

Archivum Mathematicum

We discuss the existence of almost everywhere solutions of nonlinear PDE’s of first (in the scalar and vectorial cases) and second order.

### Adjoint methods for obstacle problems and weakly coupled systems of PDE

ESAIM: Control, Optimisation and Calculus of Variations

The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton − Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived.

### An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions

Annales de l'I.H.P. Analyse non linéaire

### An existence result for a nonconvex variational problem via regularity

ESAIM: Control, Optimisation and Calculus of Variations

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The $x$-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

### An existence result for a nonconvex variational problem via regularity

ESAIM: Control, Optimisation and Calculus of Variations

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

### An explicit solution to a system of implicit differential equations

Annales de l'I.H.P. Analyse non linéaire

### An initial and boundary value problem for nonlinear composite type systems of three equations.

Mathematica Pannonica

### Approche visqueuse de solutions discontinues de systèmes hyperboliques semilinéaires

Annales de l’institut Fourier

On s’intéresse à des systèmes symétriques hyperboliques multidimensionnels en présence d’une semilinéarité. Il est bien connu que ces systèmes admettent des solutions discontinues, régulières de part et d’autre d’une hypersurface lisse caractéristique de multiplicité constante. Une telle solution ${u}^{0}$ étant donnée, on montre que ${u}^{0}$ est limite quand $\epsilon \to 0$ de solutions ${\left({u}^{\epsilon }\right)}_{\epsilon \in \right]0,1\right]}$ du système perturbé par une viscosité de taille $\epsilon$. La preuve utilise un problème mixte parabolique et des développements de couches limites....

### Blow-up time estimates for a non-local reactive-convective problem modelling sterilization of food

Banach Center Publications

### Blow-up time estimates for a non-local reactive-convective problem modelling sterilization of food

Banach Center Publications

### Boundary value problems for the stationary Vlasov-Maxwell system.

Forum mathematicum

### Convergence of an accurate scheme for first order quasi linear equations

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

### Cut and singular loci up to codimension 3

Annales de l’institut Fourier

We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension $n-2$ is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension $n-3$.

### Degenerate Eikonal equations with discontinuous refraction index

ESAIM: Control, Optimisation and Calculus of Variations

We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value...

### Degenerate stationary problems with homogeneous boundary conditions.

Electronic Journal of Differential Equations (EJDE) [electronic only]

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