Continuity of solutions of the porous media equation
Continuous dependence on modeling for some well-posed perturbations of the backward heat equation.
Continuous dependence on obstacles in double global obstacle problems.
Continuous extendibility of solutions of the Neumann problem for the Laplace equation
A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given.
Continuous extendibility of solutions of the third problem for the Laplace equation
A necessary and sufficient condition for the continuous extendibility of a solution of the third problem for the Laplace equation is given.
Control-theoretic properties of structural acoustic models with thermal effects, II. Trace regularity results
We consider a structural acoustic problem with the flexible wall modeled by a thermoelastic plate, subject to Dirichlet boundary control in the thermal component. We establish sharp regularity results for the traces of the thermal variable on the boundary in case the system is supplemented with clamped mechanical boundary conditions. These regularity estimates are most crucial for validity of the optimal control theory developed by Acquistapace et al. [Adv. Differential Equations, 2005], which ensures...
Convergence and partial regularity for weak solutions of some nonlinear elliptic equation : the supercritical case
Convergence of the coincidence set in the homogenization of the obstacle problem
Counterexample to the regularity of weak solution of elliptic systems
Counterexample to the regularity of weak solution of the quasilinear parabolic system
C?-Regularity for the Porous Medium Equation.
Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition
Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties by using assumptions...
Cross-Diffusion Systems with Entropy Structure
Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak...
Die klassisch-reguläre Lösbarkeit des Rand-Anfangswertproblems nichtlinearer Wellengleichungen mit einer Randbedingung vom gemischten Typ.
Differentiability and growth of solutions of partial differential equations.
Differentiability and partial Hölder continuity of the solutions of non-linear elliptic systems of order with quadratic growth
Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains
We consider generalized solutions to the Dirichlet problem for linear elliptic second order equations in a domain bounded by a Dini-Lyapunov surface and containing a conical point. For such solutions we derive Dini estimates for the first order generalized derivatives.
Dini-Campanato spaces and applications to nonlinear elliptic equations.
Dirichlet problem for quasi-linear elliptic equations.
Dynamics of complex singularities in 1D nonlinear parabolic PDE's
We establish local-in-time smoothing of a simple model nonlinear parabolic PDE in a scale of weighted Bergman spaces on a strip provided the weights are not too singular. This constitutes a very strong smoothing property since an immediate consequence is that the PDE can "push away" an algebraic-type complex singularity provided that the order of the singularity is small enough.