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In this paper, we investigate whether Variational Principles can be associated with the Helmholtz equation subject to impedance (absorbing) boundary conditions. This model has been extensively studied in the literature from both…
We consider the minimization of $\int_{\Omega_{\ep}} |D\vec u|^p \dd\vec x$ in a perforated domain $\Omega_{\ep}:= \Omega \setminus \bigcup_{i=1}^M B_{\ep}(\vec a_i)$ of $\R^n$, among maps $\vec u \in W^{1,p}(\Omega_{\ep}, \R^n)$ that are…
This is the first of a series of papers devoted to the initial value problem for the Euler system of compressible fluids and augmented versions containing higher-order terms. We encompass solutions that have finite total energy and enjoy a…
The linear boundary value problem under consideration describes time-harmonic motion of water in a horizontal three-dimensional layer of constant depth in the presence of an obstacle adjacent to the upper side of the layer (floating body).…
We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries of the action functional and generate…
We consider the boundedness and exponential integrability of solutions to the Dirichlet problem for the degenerate elliptic equation \[ -v^{-1}\mathrm{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=f|f|^{p-2}- v^{-1}\mathrm{Div}(v|g|^{p-2}g…
We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier-Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically…
We consider the inverse problem for the wave equation which consists of determining an unknown space-dependent force function acting on a vibrating structure from Cauchy boundary data. Since only boundary data are used as measurements, the…
A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is…
In this paper we study the shape of least-energy solutions to a singularly perturbed quasilinear problem with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that at limit, the global maximum point of…
In this paper we derive an obstacle problem with a free boundary to describe the formation of voids at areas of intense geological folding. An elastic layer is forced by overburden pressure against a V-shaped rigid obstacle. Energy…
Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. In a previous paper, we proved that any solution which is bounded in the energy space converges, along a sequence of times and in some weak sense, to a…
In this paper, we study the focusing and defocusing energy--subcritical, nonlinear wave equation in $\mathbb{R}^{1+d}$ with radial initial data for $d = 4,5$. We prove that if a solution remains bounded in the critical space on its interval…
We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation $\displaystyle -\mbox{div} (\phi(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} \Omega$…
We have recently presented an extension of the standard variational calculus to include the presence of deformed derivatives in the Lagrangian of a system of particles and in the Lagrangian density of field-theoretic models. Classical…
The Casimir effect in an inhomogeneous dielectric is investigated using Lifshitz's theory of electromagnetic vacuum energy. A permittivity function that depends continuously on one Cartesian coordinate is chosen, bounded on each side by…
The Casimir problem is usually posed as the response of a fluctuating quantum field to externally imposed boundary conditions. In reality, however, no interaction is strong enough to enforce a boundary condition on all frequencies of a…
We consider the Wulff-type energy functional $$ \mathcal{W}_\Omega(u) := \int_\Omega B(H(\nabla u (x))) - F(u(x)) \, dx, $$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this…
We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over…
We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of…