Nonlocal Robin Boundary Conditions Research Articles

Generalized nonlocal Robin Laplacian on arbitrary domains

In this paper, we prove that it is always possible to define a realization of the Laplacian \(\Delta _{\kappa ,\theta }\) on \(L^2(\Omega )\) subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of \({\mathbb {R}}^N\). This is made possible by using a capacity approach to define an admissible pair of measures \((\kappa ,\theta )\) that allows the associated form \({\mathcal {E}}_{\kappa ,\theta }\) to be closable. The nonlocal Robin Laplacian \(\Delta _{\kappa ,\theta }\) generates a sub-Markovian \(C_0\)-semigroup on \(L^2(\Omega )\) which is not dominated by the Neumann Laplacian semigroup unless the jump measure \(\theta \) vanishes.

Towards a Brezis–Oswald-type result for fractional problems with Robin boundary conditions

We consider a boundary value problem driven by the p-fractional Laplacian with nonlocal Robin boundary conditions and we provide necessary and sufficient conditions which ensure the existence of a unique positive (weak) solution. The results proved in this paper can be considered a first step towards a complete generalization of the classical result by Brezis and Oswald (Nonlinear Anal 10:55–64, 1986) to the nonlocal setting.

Diffusion with nonlocal Robin boundary conditions

We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq \beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon\partial\Omega\to{\mathscr{M}}(\overline{\Omega})$, and boundary conditions of the form $$ \partial_{\nu}^{{\mathscr{A}}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(\mathrm{d}x), \quad z\in\partial\Omega, $$ where $\partial_{\nu}^{{\mathscr{A}}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $\mu$ we show that $A_{\beta, \mu}$ generates a holomorphic semigroup $T_{\beta,\mu}$ on $L^{\infty}(\Omega)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overline{\Omega})$. Its restriction to $C(\overline{\Omega})$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.

Spectral enclosures for non-self-adjoint extensions of symmetric operators

The spectral properties of non-self-adjoint extensions A[B] of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator B. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for A[B] to have a non-empty resolvent set are provided in terms of the parameter B and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for A[B] are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with δ-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.

Long-term behavior of reaction–diffusion equations with nonlocal boundary conditions on rough domains

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Holder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction-diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.

Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $\Omega\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(\Omega)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partial\Omega)$.

The p-Laplace operator with the nonlocal Robin boundary conditions on arbitrary open sets

Let \({\Omega\subset\mathbb{R}^N}\) be an arbitrary open set with boundary \({\partial \Omega, 1 N > 1. In the first part of the article, we show that weak solutions of the quasi-linear elliptic equation \({-{\rm div}(|\nabla u|^{p-2} \nabla u)+a(x)|u|^{p-2}u=f}\) in Ω with the nonlocal Robin type boundary conditions formally given by \({|\nabla u|^{p-2} \partial u/\partial\nu+b(x)|u|^{p-2}u+\Theta_p(u)=0}\) on \({\partial \Omega}\) belong to L ∞(Ω). In the second part, assuming that Ω has a finite measure, we prove that for every \({p \in (1,\infty)}\), a realization of the operator \({\Delta_p}\) in L 2(Ω) with the above-mentioned nonlocal Robin boundary conditions generates a nonlinear order-preserving semigroup \({(S_\Theta(t))_{t \ge 0}}\) of contraction operators in L 2(Ω) if and only if \({\partial \Omega}\) is admissible (in the sense of the relative capacity) with respect to the (N − 1)-dimensional Hausdorff measure \({\fancyscript{H}^{N-1}|_{\partial \Omega}}\). We also show that this semigroup is ultracontractive in the sense that, for every \({u_0 \in L^q(\Omega)}\) (q ≥ 2) one has \({S_\Theta(t)u_0 \in L^\infty(\Omega)}\) for every t > 0. Moreover, \({\|S_\Theta(t)}\) satisfies the following (L q − L ∞)-Holder type estimate: there is a constant C ≥ 0 such that for every t > 0 and \({u_0, v_0 \in L^q(\Omega)}\) (q ≥ 2), $$\begin{array}{ll}\|S_\Theta(t)u_0-S_\Theta(t)v_0\|_{\infty, \Omega} \le C|\Omega|^\beta t^{-\delta} \|u_0-v_0\|_{q, \Omega}^\gamma,\end{array}$$ where β, δ, and γ are explicit constants depending on N, p, and q only.

Solvability of linear local and nonlocal Robin problems over [formula omitted

Let Ω ⊆ R N be a bounded Lipschitz domain. We first consider an elliptic boundary value problem with general Robin boundary conditions. The boundary conditions can be either local or nonlocal, depending on the conditions imposed on the elliptic operator. We prove that this boundary value problem is uniquely solvable, and moreover we show that such weak solution is Hölder continuous on Ω ¯ . We also prove that a realization of the associated differential operator with generalized local or nonlocal Robin boundary conditions generates an analytic C 0 -semigroup of angle π / 2 over C ( Ω ¯ ) . We conclude by applying the elliptic regularity theory to solve the corresponding Cauchy problem over C ( Ω ¯ ) .

Quasi-linear boundary value problems with generalized nonlocal boundary conditions

We investigate a quasi-linear boundary value problem of the form − div ( α | ∇ u | p − 2 ∇ u ) = 0 involving a general boundary map and mixed Neumann boundary conditions on a bounded Lipschitz domain. We show existence, uniqueness, and Hölder continuity of the weak solution of this mixed boundary value problem, and obtain maximum principles for this class of mixed equations. As a consequence, we obtain uniform continuity up to the boundary to solutions associated with a class of electrical models described by Maxwell’s equations with nonlocal boundary conditions. An extension to boundary value problems with generalized nonlocal Robin boundary conditions is also achieved.

The one-dimensional wave equation with general boundary conditions

We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions α j u′(j) + β j u(j) + γ 1–j u(1 − j) = 0, (j = 0, 1) generates a cosine family on L p (0, 1) for every $${p\,{\in}\,[1,\infty)}$$ . Here α j , β j and γ j are complex numbers satisfying α 0, α 1 ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p ∈ ( 1 , N ) , Ω ⊂ R N a bounded W 1 , p -extension domain and let μ be an upper d-Ahlfors measure on ∂ Ω with d ∈ ( N − p , N ) . We show in the first part that for every p ∈ [ 2 N / ( N + 2 ) , N ) ∩ ( 1 , N ) , a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L 2 ( Ω ) , and hence, the associated first order Cauchy problem is well posed on L q ( Ω ) for every q ∈ [ 1 , ∞ ) . In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.