We prove the local existence of solutions for equations of motion of a viscous compressible barotropic fluid in a domain bounded by a free surface. The solutions are shown to exist in exactly those function spaces where global solutions were found in our previous papers [14, 15].
We study the existence of principal eigenvalues for differential operators of second order which are not necessarily in divergence form. We obtain results concerning multiplicity of principal eigenvalues in both the variational and the general case. Our approach uses systematically the Krein-Rutman theorem and fixed point arguments for the spectral radius of some associated problems. We also use a variational characterization for both the self-adjoint and the general case.
In this paper, we consider probability measures μ and ν on a d-dimensional sphere in and cost functions of the form that generalize those arising in geometric optics where We prove that if μ and ν vanish on -rectifiable sets, if |l'(t)|>0, and is monotone then there exists a unique optimal map To that transports μ onto where optimality is measured against c. Furthermore, Our approach is based on direct variational arguments. In the special case when existence of optimal maps...
In this paper, we are concerned with the semilinear parabolic equation ∂u/∂t - Δu = g(t,x,u) if u = 0 if , where is a bounded domain with smooth boundary ∂Ω and is T-periodic with respect to the first variable. The existence and the multiplicity of T-periodic solutions for this problem are shown when g(t,x,ξ)/ξ lies between two higher eigenvalues of - Δ in Ω with the Dirichlet boundary condition as ξ → ±∞.