Marius Mitrea

Stability results on interpolation scales of quasi-Banach spaces and applications

We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE’s and several applications are presented.

Clifford wavelets, singular integrals, and Hardy spaces

Clifford algebras.- Constructions of Clifford wavelets.- The L 2 Boundedness of Clifford algebra valued singular integral operators.- Hardy spaces of monogenic functions.- Applications to the theory of harmonic functions.

Clifford algebras and Maxwell's equations in Lipschitz domains

We present a simple, Clifford algebra-based approach to several key results in the theory of Maxwells equations in non-smooth subdomains of R m . Among other things, we give new proofs to the boundary energy estimates of Rellich type for Maxwells equations in Lipschitz domains from [20, 10], discuss radiation conditions and the case of variable wave number.

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

Abstract We study spectral properties for H K , Ω , the Krein–von Neumann extension of the perturbed Laplacian − Δ + V defined on C 0 ∞ ( Ω ) , where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ R n belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C 1 , r , r > 1 / 2 . In particular, in the aforementioned context we establish the Weyl asymptotic formula # { j ∈ N | λ K , Ω , j ⩽ λ } = ( 2 π ) − n v n | Ω | λ n / 2 + O ( λ ( n − ( 1 / 2 ) ) / 2 ) as λ → ∞ , where v n = π n / 2 / Γ ( ( n / 2 ) + 1 ) denotes the volume of the unit ball in R n , and λ K , Ω , j , j ∈ N , are the non-zero eigenvalues of H K , Ω , listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − Δ + V defined on C 0 ∞ ( Ω ) ) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980s. Our work builds on that of Grubb in the early 1980s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exterior-type domains Ω = R n ∖ K , n ⩾ 3 , with K ⊂ R n compact and vanishing Bessel capacity B 2 , 2 ( K ) = 0 , to prove equality of Friedrichs and Krein Laplacians in L 2 ( Ω ; d n x ) , that is, − Δ | C 0 ∞ ( Ω ) has a unique nonnegative self-adjoint extension in L 2 ( Ω ; d n x ) .

The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that if 0<p<1, 0<q<1, ℝ, while if 0<p<1, 1≤q<+∞, ∝, and if 1≤p<+∞, 0<q<1, ℝ.Applications to questions regarding the global interior regularity of solutions to Poisson type problems for the three-dimensional Lamé system in Lipschitz domains are presented.

Robin-to-Robin Maps and Krein-Type Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains

We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains in ℝn, n ⩾ 2, with generalized Robin boundary conditions.

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.

Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds

We solve three basic potential theoretic problems: Hodge decompositions for vector fields, Poisson problems for the Hodge-Laplacian, and inhomogeneous Maxwell equations, in arbitrary Lipschitz subdomains of a smooth, compact, three dimensional, Riemannian manifold. In each case we derive sharp estimates on Sobolev-Besov scales and establish integral representation formulas for the solution. The proofs rely on tools from harmonic analysis and algebraic topology, such as Calderon-Zygmund theory and the de Rham theory.

Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains

In the first part of this article we give intrinsic characterizations of the classes of Lipschitz and C1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C1. In the second part of the article, we study the invariance of various classes of domains of locally finite perimeter under bi-Lipschitz and C1 diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C1 diffeomorphisms. A number of other applications are also presented.

On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds

We study the analyticity of the semigroup generated by the Stokes operator equipped with Neumann-type boundary conditions on L P spaces in Lipschitz domains. Our strategy is to regularize this operator by considering the Hodge Laplacian, which has the additional property that it commutes with the Leray projection.