An evolutionary double-well problem
An example of a nonlinear second order elliptic system in three dimension
We provide an explicit example of a nonlinear second order elliptic system of two equations in three dimension to compare two -regularity theories. We show that, for certain range of parameters, the theory developed in Daněček, Nonlinear Differential Equations Appl.9 (2002), gives a stronger result than the theory introduced in Koshelev, Lecture Notes in Mathematics,1614, 1995. In addition, there is a range of parameters where the first theory gives H"older continuity of solution for all , while...
An example of irregular solution to a nonlinear Euler-Lagrange elliptic system with real analytic coefficients
An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit
The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on . This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss the general low...
An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit
The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss...
An example of the absence of a global solution to some inverse problem for a hyperbolic equation.
An example of the Heisenberg group
An existence and approximation theorem for solutions of degenerate quasilinear elliptic equations
The main result establishes that a weak solution of degenerate quasilinear elliptic equations can be approximated by a sequence of solutions for non-degenerate quasilinear elliptic equations.
An existence and localization theorem for the solutions of a Dirichlet problem
We establish an existence theorem for a Dirichlet problem with homogeneous boundary conditions by using a general variational principle of Ricceri.
An existence criterion for -norms of higher-order derivatives of solutions to a homogeneous parabolic equation.
An existence proof for the stationary compressible Stokes problem
In this paper, we prove the existence of a solution for a quite general stationary compressible Stokes problem including, in particular, gravity effects. The Equation Of State gives the pressure as an increasing superlinear function of the density. This existence result is obtained by passing to the limit on the solution of a viscous approximation of the continuity equation.
An existence result for a free boundary problem for the -Laplace operator.
An existence result for a nonconvex variational problem via regularity
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 -dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.
An existence result for a nonconvex variational problem via regularity
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 quadrature surface free boundary problem
The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.
An existence result for a semipositone problem with a sign changing weight.
An existence result for an interior electromagnetic casting problem
This paper deals with an interior electromagnetic casting (free boundary) problem. We begin by showing that the associated shape optimization problem has a solution which is of class C 2. Then, using the shape derivative and the maximum principle, we give a sufficient condition that the minimum obtained solves our problem.
An existence result for balance laws with multifunctions: a model from the theory of granular media
We study an example of the balance law with a multifunction source term, coming from the theory of granular media. We prove the existence of "weak entropy solutions" to this system, using the vanishing viscosity method and compensated compactness. Because of the occurrence of a multifunction we give a new definition of the weak entropy solutions.
An existence result for elliptic problems with singular critical growth.
An existence result for nonlinear elliptic problems involving critical Sobolev exponent