DCDS

Let $Ω\subset\mathbb{R}^N$ ($N\ge 2$) be a bounded domain with a boundary $∂Ω$ of class $C^2$ and let $\alpha,\beta$ be maximal monotone graphs in $\mathbb{R}^2$ satisfying $\alpha(0)\cap\beta(0)\ni 0$. Given $f\in L^1(Ω)$ and $g\in L^1(∂Ω)$, we characterize the existence and uniqueness of weak solutions to the semi-linear elliptic equation $-\Delta u+\alpha(u)\ni f$ in $Ω$ with the nonlinear general Wentzell boundary conditions $-\Delta_{\Gamma} u+\frac{\partial u}{\partial\nu}+\beta(u)\ni g$ on $∂Ω$. We also show the well-posedness of the associated parabolic problem on the Banach space $L^1(Ω)\times L^1(∂Ω)$.

DCDS

We study the interior approximate controllability of fractional wave equations with the fractional Caputo derivative associated with a non-negative self-adjoint operator satisfying the unique continuation property. Some well-posedness and fine regularity properties of solutions to fractional wave and fractional backward wave type equations are also obtained. As an example of applications of our results we obtain that if $1<\alpha<2$ and $\Omega\subset\mathbb{R}^N$ is a smooth connected open set with boundary $\partial\Omega$, then the system $\mathbb D_t^\alpha u+A_Bu=f$ in $\Omega\times (0,T)$, $u(\cdot,0)=u_0$, $\partial_tu(\cdot,0)=u_1$, is approximately controllable for any $T>0$, $(u_0,u_1)\in V_{\frac{1}{\alpha}}\times L^2(\Omega)$, $\omega\subset\Omega$ any open set and any $f\in C_0^\infty(\omega\times (0,T))$. Here, $A_B$ can be the realization in $L^2(\Omega)$ of a symmetric non-negative uniformly elliptic operator with Dirichlet or Robin boundary conditions, or the realization in $L^2(\Omega)$ of the fractional Laplace operator $(-\Delta)^s$ ($0< s <1$) with the Dirichlet boundary condition ($u=0$ on $\mathbb{R}^N\setminus\Omega$) and the space $V_{\frac{1}{\alpha}}$ denotes the domain of the fractional power of order $\frac{1}{\alpha}$ of the operator $A_B$.

EECT

We investigate a class of semilinear parabolic and elliptic problems with
fractional dynamic boundary conditions. We introduce two new operators, the
so-called fractional Wentzell Laplacian and the fractional Steklov operator,
which become essential in our study of these nonlinear problems. Besides
giving a complete characterization of well-posedness and regularity of
bounded solutions, we also establish the existence of finite-dimensional
global attractors and also derive basic conditions for blow-up.

DCDS

We investigate the long term behavior in terms of finite dimensional global
attractors and (global) asymptotic stabilization to steady states, as time
goes to infinity, of solutions to a non-local semilinear reaction-diffusion
equation associated with the fractional Laplace operator on non-smooth
domains subject to Dirichlet, fractional Neumann and Robin boundary
conditions.

DCDS

Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients.
We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.

CPAA

Let $\Omega\subset R^N$ be a bounded open set with Lipschitz continuous boundary $\partial \Omega$. We define a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on $L^2(\partial \Omega)$ which can also be ultracontractive. We also use the fractional Dirichlet-to-Neumann operator to compare the eigenvalues of a realization in $L^2(\Omega)$ of the fractional Laplace operator with Dirichlet boundary condition and the regional fractional Laplacian with the fractional Neumann boundary conditions.

CPAA

We show that on a bounded domain $\Omega\subset R^N$ with Lipschitz continuous boundary $\partial \Omega$, weak solutions of the elliptic equation $\lambda u-Au=f$ in $\Omega$ with the boundary conditions $-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ are globally Hölder continuous on $\bar \Omega$. Here $A$ is a uniformly elliptic operator in divergence form with bounded measurable coefficients, $\Delta_\Gamma$ is the Laplace-Beltrami operator on $\partial \Omega$, $\partial_\nu^a u$ denotes the conormal derivative of $u$, $\lambda,\gamma>0$ are real numbers and $\beta$ is a bounded measurable function on $\partial Omega$. We also obtain that a realization of the operator $A$ in $C(\bar \Omega)$ with the general Wentzell boundary conditions $(Au)|_{\partial \Omega}-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ generates a strongly continuous compact semigroup. Some analyticity results of the semigroup are also discussed.

MCRF

In this paper we study optimal control problems with the regional fractional $p$-Laplace equation, of order $s \in \left( {0,1} \right)$ and $p \in \left[ {2,\infty } \right)$, as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the regional fractional $p$-Laplace operator. We show existence and uniqueness of solutions to the state equations and existence of solutions to the optimal control problems. We prove that the regional fractional $p$-Laplacian approaches the standard $p$-Laplacian as $s$ approaches 1. In this sense, this fractional $p$-Laplacian can be considered degenerate like the standard $p$-Laplacian. To overcome this degeneracy, we introduce a regularization for the regional fractional $p$-Laplacian. We show existence and uniqueness of solutions to the regularized state equation and existence of solutions to the regularized optimal control problem. We also prove several auxiliary results for the regularized problem which are of independent interest. We conclude with the convergence of the regularized solutions.