Irina Mitrea

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

MIXED BOUNDARY VALUE PROBLEMS FOR THE STOKES SYSTEM

We prove the well-posedness of the mixed problem for the Stokes system in a class of Lipschitz domains in ℝ n , n ≥ 3. The strategy is to reduce the original problem to a boundary integral equation, and we establish certain new Rellich-type estimates which imply that the intervening boundary integral operator is semi-Fredholm. We then prove that its index is zero by performing a homotopic deformation of it onto an operator related to the Lame system, which has recently been shown to be invertible.

A remark on the regularity of the div-curl system

As a limiting case of the classical Calderon-Zygmund theory, in this note we study the Besov regularity of differential forms u for which du and δu have absolutely integrable coefficients in R n .

On the Regularity of Green Functions in Lipschitz Domains

In this paper we study regularity properties of Green functions associated with elliptic differential operators L in Lipschitz domains. In particular, we discuss the membership of G D (x, ·) and to weak Sobolev spaces in bounded Lipschitz domains Ω, uniformly for x ∈ Ω, where G D is the Green function with Dirichlet boundary condition associated with L, δ∂Ω is the distance function to the boundary of Ω, and α ∈ [0, 1]. Our analysis includes the case of second and higher order elliptic systems with constant coefficients, the bi-Laplacian, as well as the Stokes system.

On the Besov regularity of conformal maps and layer potentials on nonsmooth domains

Abstract We study the Besov regularity of conformal mappings for domains with rough boundary based on the well-posedness for the Dirichlet problem with Besov data. Also, sharp invertibility results for the classical layer potential operators on Sobolev–Besov spaces on the boundary of curvilinear polygons are obtained.

Transmission boundary problems for Dirac operators on Lipschitz domains and applications to Maxwell’s and Helmholtz’s equations

The transmission boundary value problem for a perturbed Dirac operator on arbitrary bounded Lipschitz domains in R3 is formulated and solved in terms of layer potentials of Clifford-Cauchy type. As a byproduct of this analysis, an elliptization procedure for the Maxwell system is devised which allows us to show that the Maxwell and Helmholtz transmission boundary value problems are well-posed as a corollary of the unique solvability of this more general Dirac transmission problem.

Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains

By producing a L2 convergent Neumann series, we prove the invertibility of the elastostatics and hydrostatics boundary layer potentials on arbitrary Lipschitz domains with small Lipschitz character and 3D polyhedra with large dihedral angles.

Groupoid Metrization Theory

The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complet, detailed proofs, and a large number of examples and counterexamples are provided.

Boundary value problems and integral operators for the bi-Laplacian in non-smooth domains

— In this paper we explore the e¤ectiveness of the classical method of layer potentials in the treatment of boundary value problems for the bi-Laplacian formulated in arbitrary Lipschitz domains, Lipschitz domains whose outward unit normal has small mean-oscillations, and domains of class C.

Sobolev Estimates for the Green Potential Associated with the Robin—Laplacian in Lipschitz Domains Satisfying a Uniform Exterior Ball Condition

We show that if u = Gλf is the solution operator for the Robin problem for the Laplacian, i.e., Δu = f in Ω, ∂ νu + λu = 0 on ∂Ω (with 0 ≤ λ ≤ ∞), then Gλ : Lp(Ω) → W 2,p(Ω) is bounded if 1 < p ≤ 2 and Ω ⊂ ℝn is a bounded Lipschitz domain satisfying a uniform exterior ball condition. This extends the earlier results of V. Adolfsson, B. Dahlberg, S. Fromm, D. Jerison, G. Verchota, and T. Wolff, who have dealt with Dirich-let (λ = ∞) and Neumann (λ = 0) boundary conditions. Our treatment of the end-point case p = 1 works for arbitrary Lipschitz domains and is conceptually different from the proof given by the aforementioned authors.