DCDS
Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data
Paul Sacks Mahamadi Warma
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 761-787 doi: 10.3934/dcds.2014.34.761
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 $L^1(Ω)\times L^1(∂Ω)$.
keywords: existence of weak solutions Semi-linear elliptic equations nonlinear Wentzell boundary conditions elliptic-parabolic equations mild solutions.
DCDS
On the interior approximate controllability for fractional wave equations
Valentin Keyantuo Mahamadi Warma
Discrete & Continuous Dynamical Systems - A 2016, 36(7): 3719-3739 doi: 10.3934/dcds.2016.36.3719
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 $A_B$.
keywords: existence and regularity of solutions Fractional wave equations interior approximate controllability.
EECT
Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions
Ciprian G. Gal Mahamadi Warma
Evolution Equations & Control Theory 2016, 5(1): 61-103 doi: 10.3934/eect.2016.5.61
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.
keywords: elliptic problem fractional Wentzell boundary conditions The fractional Laplace operator fractional Dirichlet-to-Neumann operator. exponential attractor global attractor fractional Steklov operator semilinear reaction-diffusion equation
DCDS
Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions
Ciprian G. Gal Mahamadi Warma
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1279-1319 doi: 10.3934/dcds.2016.36.1279
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.
keywords: global attractor asymptotic behavior. convergence to steady states The fractional Laplace operator semi-linear reaction-diffusion equation fractional Neumann and Robin boundary conditions on open sets
DCDS
Semi linear parabolic equations with nonlinear general Wentzell boundary conditions
Mahamadi Warma
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5493-5506 doi: 10.3934/dcds.2013.33.5493
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.
keywords: Second order elliptic operators in divergence form nonlinear general Wentzell boundary conditions nonlinear semigroup of contractions.
CPAA
A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains
Mahamadi Warma
Communications on Pure & Applied Analysis 2015, 14(5): 2043-2067 doi: 10.3934/cpaa.2015.14.2043
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.
keywords: Fractional Laplacian semigroup. fractional Dirichlet-to-Neumann operator form methods
CPAA
Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains
Mahamadi Warma
Communications on Pure & Applied Analysis 2013, 12(5): 1881-1905 doi: 10.3934/cpaa.2013.12.1881
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.
keywords: analytic semigroups. general Wentzell boundary conditions weak solution Schauder estimates Second order elliptic operators Hölder continuity
MCRF
Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence
Harbir Antil Mahamadi Warma
Mathematical Control & Related Fields 2018, 8(0): 1-38 doi: 10.3934/mcrf.2019001

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.

keywords: Regional fractional p-Laplace operator non-constant coefficient quasi-linear nonlocal elliptic boundary value problems optimal control

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