We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton-Jacobi Equation ut−∆u+|Du| m = f in Ω×(0,+∞) with inhomogeneous Dirichlet boundary conditions, where Ω is a bounded domain of RN . We mainly focus on the superquadratic case (m > 2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f, the initial...

We consider the Navier–Stokes–Fourier system describing motion of a compressible, viscous, and heat-conducting fluid in bounded domain Ω⊂Rd, d = 2, 3, with general non-homogeneous Dirichlet boundary conditions for velocity absolute temperature, associated density on inflow part. introduce new concept weak solution based satisfaction entropy inequality together balance law ballistic energy. show...

We consider a controlled state equation of parabolic type on the halfline (0,+∞) with boundary conditions of Dirichlet type in which the unknown is equal to the sum of the control and of a white noise in time. We study finite horizon and infinite horizon optimal control problem related by menas of backward stochastic differential equations.

Abstract. This paper is companion to our earlier work [8] (see also announcement [7]). Let M be a closed manifold and Y be an embedded hypersurface, such that there is a decomposition of M = M1 ∪M2 into two manifolds with boundary M1 and M2 , with M1 ∩M2 = Y . In [8] we proved the decomposition formula for detζ∆ the ζ-determinant of a Dirac Laplacian ∆ on M . The contributions coming from M1 an...

Abstract We study the first eigenvalue of the Laplace equation in a bounded domain in R (d = 2, 3) with mixed Neumann–Dirichlet (Zaremba) boundary conditions. The Neumann condition is imposed on most of the boundary and the Dirichlet boundary consists of a cluster of small windows. When the windows are well separated the first eigenvalue is asymptotically the sum of eigenvalues of mixed problem...

We investigate the bifurcation structure of the Kuramoto-Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and O(2) symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary ...