We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for ...triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.
This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space ...of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.
Deepening the study of a new approximation sequence of positive linear operators we introduced and studied in 12, in this paper we disclose its relationship with the Markov semigroup (pre)generation ...problem for a class of degenerate second-order elliptic differential operators which naturally arise through an asymptotic formula, as well as with the approximation of the relevant Markov semigroups in terms of the approximating operators themselves.
The analysis is carried out in the context of the space C(K) of all continuous functions defined on an arbitrary convex compact subset K of Rd, d≥1, having non-empty interior and a not necessarily smooth boundary, as well as, in some particular cases, in Lp(K) spaces, 1≤p<+∞. The approximation formula also allows to infer some preservation properties of the semigroup such as the preservation of the Lipschitz-continuity as well as of the convexity. We finally apply the main results to some noteworthy particular settings such as balls and ellipsoids, the unit interval and multidimensional hypercubes and simplices. In these settings the relevant differential operators fall into the class of Fleming–Viot operators.
On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients ...in the boundary conditions are functions. We prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators, provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty, open set.
For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an ...arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expression.
Let $\Omega$ be a bounded domain in $R^{n}$ with smooth boundary $\partial\Omega$. In this article, we will investigate the spectral properties of a non-self adjoint elliptic differential ...operator\\ $(Au)(x)=-\sum^{n}_{i,j=1}\left(\omega^{2\alpha}(x)a_{ij}(x) \mu(x)u'_{x_{i}}(x)\right)'_{x_{j}}$, acting in the Hilbert space $H=L^{2}{(\Omega)}$. with Dirichlet-type boundary conditions. Here $a_{ij}(x)= \overline{a_{ji}(x)}\;\;\;(i,j=1,\ldots,n),\;\;\; a_{ij}(x)\in C^{2}(\overline{\Omega})$, and the functions $a_{ij}(x)$ satisfies the uniformly elliptic condition, and let $ 0 \leq \alpha < 1$. Furthermore, for $\forall x \in \overline{\Omega}$, the function $\mu(x)$ lie in the $\psi_{\theta_1\theta_2}$ , where ${\psi_{\theta_1\theta_2}}=\{z \in {\bf C}:\;\pi/2<\theta_1 \leq|arg\;z| \leq \theta_2<\pi\},$
The spectrum of a selfadjoint second order elliptic differential operator in L2(Rn) is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional ...Glazman decomposition and correspond to an interior and an exterior boundary value problem. This leads to PDE analogs of renowned facts in spectral theory of ODEs. The main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding Weyl functions, and are applied to the PDE setting afterwards.
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