On non-coercive mixed problems for parameter-dependent elliptic operators

We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain $D$ of ${\mathbb R}^n$ for a second order parameter-dependent elliptic differential operator $A (x,\partial, \lambda)$ with complex-valued essentially bounded measured coefficients and complex parameter $\lambda$. The differential operator is assumed to be of divergent form in $D$, the boundary operator $B (x,\partial)$ is of Robin type with possible pseudo-differential components on $\partial D$. The boundary of $D$ is assumed to be a Lipschitz surface. Under these assumptions the pair $(A (x,\partial, \lambda),B)$ induces a holomorphic family of Fredholm operators $L(\lambda): H^+(D) \to H^- (D)$ in suitable Hilbert spaces $H^+(D)$ , $H^- (D)$ of Sobolev type. If the argument of the complex-valued multiplier of the parame\-ter in $A (x,\partial, \lambda)$ is continuous and the coefficients related to second order derivatives of the operator are smooth then we prove that the operators $L(\lambda)$ are conti\-nu\-ously invertible for all $\lambda$ with sufficiently large modulus $|\lambda|$ on each ray on the complex plane $\mathbb C$ where the differential operator $A (x,\partial, \lambda)$ is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family $L (\lambda)$ to be (doubly) complete in the spaces $H^+(D)$, $H^- (D)$ and the Lebesgue space $L^2 (D)$

On a mixed problem for the parabolic Lam'e type operator

We consider a boundary value problem for the parabolic Lam\'e type operator being a linearization of the Navier-Stokes' equations for compressible flow of Newtonian fluids. It consists of recovering a vector-function, satisfying the parabolic Lam\'e type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding H\"older spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the Integral Representation's Method we obtain the Uniqueness Theorem and solvability conditions for the problem

On Runge type theorems for solutions to strongly uniformly parabolic operators

Let $G_1, G_2$ be domains in ${\mathbb R}^{n+1}$, $n \geq 2$, such that $G_1 \subset G_2$ and the domain $G_1$ have rather regular boundary. We investigate the problem of approximation of solutions to strongly uniformly $2m$-parabolic system $\mathcal L$ in the domain $G_1$ by solutions to the same system in the domain $G_2$. First, we prove that the space $S _{\mathcal L}(G_2)$ of solutions to the system $\mathcal L$ in the domain $G_2$ is dense in the space $S _{\mathcal L}(G_1)$, endowed with the standard Fr\'echet topology of the uniform convergence on compact subsets in $G_1$, if and only if the complements $G_2 (t) \setminus G_1 (t)$ have no non-empty compact components in $G_2 (t)$ for each $t\in \mathbb R$, where $G_j (t) = \{x \in {\mathbb R}^n: (x,t) \in G_j\}$. Next, under additional assumptions on the regularity of the bounded domains $G_1$ and $G_1(t)$, we prove that solutions from the Lebesgue class $L^2(G_1)\cap S _{\mathcal L}(G_1)$ can be approximated by solutions from $S _{\mathcal L}(G_2)$ if and only if the same assumption on the complements $G_2 (t) \setminus G_1 (t)$, $t\in \mathbb R$, is fulfilled

Negative Sobolev Spaces in the Cauchy Problem for the Cauchy-Riemann Operator

Let D be a bounded domain in Cn (n > 1) with a smooth boundary @D. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for the Cauchy-Riemann operator @ in D. In particular, we describe traces of the corresponding Sobolev functions on @D and give an adequate formulation of the problem. Then we prove the uniqueness theorem for the problem, describe its necessary and sufficient solvability conditions and produce a formula for its exact solution

STURM-LIOUVILLE PROBLEMS IN WEIGHTED SPACES OVER DOMAINS WITH NON-SMOOTH EDGES. I

We consider (in general noncoercive) mixed problems in a bounded domain D in Rn for a second-order elliptic partial differential operator A(x, ∂). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ∂) is the restriction of a linear combination of the function and its derivatives to ∂D and the boundary of D is a Lipschitz surface. We separate a closed set Y ⊂ ∂D and control the growth of solutions near Y . We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y . Finally, we prove the completeness of the root functions associated with L. The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskiı˘ spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms

Boundary Problems for Helmholtz Equation and the Cauchy Problem for Dirac Operators

Studying an operator equation Au = f in Hilbert spaces one usually needs the adjoint operator A* for A. Solving the ill-posed Cauchy problem for Dirac type systems in the Lebesgue spaces by an iteration method we propose to construct the corresponding adjoint operator with the use of normally solvable mixed problem for Helmholtz Equation. This leads to the description of necessary and sufficient solvability conditions for the Cauchy Problem and formulae for its exact and approximate solutionsПри изучении операторного уравнения Au = f в пространствах Гильберта обычно требуется знать сопряженный A* оператор для A. Решая некорректную задачу Коши для операторов типа Дирака в пространствах Лебега одним итерационным методом, мы предлагаем построить соответствующий сопряженный оператор при помощи нормально разрешимой смешанной задачи для уравнения Гельмгольца. Это ведет к описанию условий разрешимости задачи Коши и к построению ее точного и приближенных решений