Related papers: Singular limit problem of abstract second order ev…
We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or…
This article is concerned with two inverse problems on determining moving source profile functions in evolution equations with a derivative order $\alpha\in(0,2]$ in time. In the first problem, the sources are supposed to move along known…
A stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space is considered. Thanks to the identification of the mild solution of the state equation as $\nu$-weak Dirichlet process, the value…
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…
We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time $t_0$, where…
In this paper we deal with a second order nonlinear evolution inclusion, with a nonmonotone, noncoercive viscosity term. Using a parabolic regularization (approximation) of the problem and {\it a priori} bounds that permit passing to the…
In this paper we consider the higher order Lioville-type equation $(-\Delta)^{m} u=\rho^{2m} V(x) e^{u}$ in $\Omega\subseteq\mathbb{R}^{2m}$ with $V\neq0$ a given smooth potential, $\rho\in\mathbb{R}^{+}$ a small parameter which tends to…
Direct and inverse initial-boundary problems on a bounded interval for systems of quasilinear evolution equations with general nonlinearities are considered. In the case of inverse problems conditions of integral overdetermination are…
We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with…
We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and insufficient regularity, where the equation in the limit with noise would not be defined due to lack of regularity. We…
We consider singular perturbation elliptic problems depending on a parameter ? such that, for ? = 0 the boundary conditions are not adapted to the equation (they do not satisfy the Shapiro - Lopatinskii condition). The limit only holds in…
We consider a Dirichlet elliptic problem driven by the Laplacian with singular and superlinear nonlinearities. The singular term appears on the left-hand side while the superlinear perturbation is parametric with parameter $\lambda>0$ and…
Boundary value problem for complete second order elliptic equation is considered in Banach space. The equation and boundary conditions involve a small and spectral parameter. The uniform L_{p}-regularity properties with respect to space…
In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a variational problem with constraints. Then, our abstract formulation is applied to study several optimization setups including closed CMC…
Optimal Dirichlet boundary control for a fractional/normal evolution with a final observation is considered. The unique existence of the solution and the first-order optimality condition of the optimal control problem are derived. The…
Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on…
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the…
In this paper we consider second order evolution equations with bounded damping. We give a characterization of a non uniform decay for the damped problem using a kind of observability estimate for the associated undamped problem.
We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and…
We characterize the behavior of the solutions of linear evolution partial differential equations on the half line in the presence of discontinuous initial conditions or discontinuous boundary conditions, as well as the behavior of the…