Related papers: A Shape Design Approximation for Degenerate Partia…

In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and…

This work addresses an inverse problem for a semi-discrete parabolic equation, consisting of identifying the right-hand side of the equation from solution measurements at an intermediate time and within a spatial subdomain. We apply this…

We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension…

We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions…

While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology…

We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H\"older semi-norms not with respect to all, but only with respect to some of the independent variables.

We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a…

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and…

We study the boundary regularity properties and derive a priori pointwise supremum estimates of weak solutions and their derivatives in terms of suitable weighted $L^2$-norms for a class of degenerate parabolic equations that satisfy…

We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions…

In this paper we consider a degenerate population equation in divergence form depending on time, on age and on space and we prove a related null controllability result via Carleman estimates.

In this paper, we analyze the null controllability property for a degenerate parabolic equation involving memory terms with a locally distributed control. We first derive a null controllability result for a nonhomogeneous degenerate heat…

This paper is concerned with the null controllability for linear backward stochastic parabolic equations with dynamic boundary conditions and convection terms. Using the classical duality argument, the null controllability is obtained via…

The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method builds upon the formulation introduced in Bertalmio et al., J. Comput. Phys., 174 (2001),…

Convergence is proven for Schwarz-like methods applied to degenerate elliptic-parabolic equations with a $p$-structure. This family of PDEs, e.g., arises when modelling nonlinear diffusion processes. The Schwarz-like approximation methods…

We investigate a backward anisotropic stochastic parabolic equation with general dynamic boundary conditions, where the drift involves both $\mathbb{L}^2$ and $\mathbb{H}^{-1}$ bulk--surface terms. We first establish the well-posedness of…

This article deals with the boundary null controllability of some degenerate parabolic equations posed on a square domain, presenting the first study of boundary controllability for such equations in multidimensional settings. The proof…

Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine…

This paper considers a semi-discrete forward stochastic parabolic operator with homogeneous Dirichlet conditions in arbitrary dimensions. We show the lack of null controllability for a spatial semi-discretization of a null-controllable…

This thesis pertains to the study of elliptic and parabolic partial differential equations on "thin" structures. The first main objective is to establish the strong and weak low-dimensional counterparts of the parabolic Neumann problem. The…