Lubomira G. Softova

Elliptic and Parabolic Equations with Discontinuous Coefficients

Introduction. Boundary Value Problems for Linear Operators with Discontinuous Coefficients. Linear and Quasilinear Operators with VMO Coefficients. Nonlinear Operators with Discontinuous Coefficients. Appendix A1: Functional and Real Analysis Tools. Appendix A2: Maximum Principles. Bibliography. Index. Functional Spaces and Their Respective Norms.

Parabolic systems with measurable coefficients in weighted Orlicz spaces

We consider a parabolic system in divergence form with measurable coefficients in a cylindrical space–time domain with nonsmooth base. The associated nonhomogeneous term is assumed to belong to a suitable weighted Orlicz space. Under possibly optimal assumptions on the coefficients and minimal geometric requirements on the boundary of the underlying domain, we generalize the Calderon–Zygmund theorem for such systems by essentially proving that the spatial gradient of the weak solution gains the same weighted Orlicz integrability as the nonhomogeneous term.

Parabolic oblique derivative problem with discontinuous coefficients in generalized Morrey spaces

We study the regularity of the solutions of the oblique derivative problem for linear uniformly parabolic equations with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized Morrey space than the strong solution belongs to the corresponding generalized Sobolev-Morrey space.

The Dirichlet problem for elliptic equations with VMO coefficients in generalized Morrey spaces

We consider the Dirichlet problem in a bounded smooth domain \( \Omega \subset \mathbb{R}^{n} \) for linear uniformly elliptic equation \( \mathfrak{L}u(x)=f(x) \) with VMO principal coefficients. Its unique strong solvability is proved in [5] and [6]. Our aim is to show that for every f belonging to the generalized Morrey space \( L^{p,\omega}(\Omega),p \in (1,\infty),\omega:\mathbb{R}^{n}\times\mathbb{R}_{+}\rightarrow \mathbb{R}_{+} \rm {the\; operator} \mathfrak{L}:W^{2,p,\omega}\cap W_{0}^{1,p}(\Omega)\rightarrow L^{p,\omega}(\Omega) \) is bijective and the estimate \( \parallel D^{2}u \parallel _{L^{p,\omega}(\Omega)}\leq C(\parallel {f} \parallel_{L^{p,\omega}(\Omega)}+ \parallel{u}\parallel_ {L^{p,\omega}(\Omega)}) \) holds.

Parabolic oblique derivative problem in generalized Morrey spaces

We study the regularity of the solutions of the oblique derivative problem for linear uniformly parabolic equations with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized Morrey space than the strong solution belongs to the corresponding generalized Sobolev-Morrey space.

Parabolic Obstacle Problem with Measurable Coefficients in Morrey-Type Spaces

We consider obstacle problem related to linear divergence form parabolic system with measurable coefficients in domain with irregular boundary. Supposing that the data of the problem and the obstacle belong to Morrey-type space, we get Calderon–Zygmund type estimate for the gradient of the solution.

Applications of Differential Calculus to Nonlinear Elliptic Boundary Value Problems with Discontinuous Coefficients

We deal with Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. First we state suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Then we fix a solution u0 such that the linearized in u0 problem is non-degenerate, and we apply the Implicit Function Theorem: For all small perturbations of the coefficient functions there exists exactly one solution u ≈ u0, and u depends smoothly (in W 2,p with p larger than the space dimension) on the data. For that no structure and growth conditions are needed, and the perturbations of the coefficient functions can be general L∞-functions with respect to the space variable x. Moreover we show that the Newton Iteration Procedure can be applied to calculate a sequence of approximate (in W 2,p again) solutions for u0.