Fractional elliptic equations, Caccioppoli estimates and regularity

By Luis Caffarelli, Pablo Raúl Stinga. 2014

269 citations recorded by the source corpus.

Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\ u=0,&\hbox{on}~\partial\Omega, \end{cases}\quad \hbox{and}\quad \begin{cases} L^su=f,&\hbox{in}~\Omega,\\ \partial_Au=0,&\hbox{on}~\partial\Omega. \end{cases}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and $\partial_Au$ is the conormal derivative of $u$ with respect to the coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients $A(x)$, the right hand side $f$ and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman--Stampacchia--Weinberger and we obtain nonlocal integro-differential formulas for $L^su(x)$. Essential tools in the analysis are the semigroup language approach and the extension problem.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/86894cc6ce69668e6e6d44ab2e0ad2ffcfa614e9

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Citations in the corpus

Nonlocal Diffusion and Applications (2015, methodology): “This technique is classical (see [92]), but it has also been efficiently used in recent research papers (see for instance [34, 90, 22]).”
Quantitative approximation properties for the fractional heat equation (2017, background): “work. Let L= @ n+1x 1 2s n+1 @ n+1 + x 1 2s n+1 @ ia ij@ j, where i;j2f1;:::;ngand aij : Rn !R n sym is a positive denite, symmetric, Lipschitz continuous matrix eld. Then, following [CS07], [ST10], [CS16], we dene ( aij) su(x0) := lim x n+1!0 x1 2s n+1@ u(x 0;x ); where usolves the equation Lu = 0 ”
Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator (2016, background): “We also refer to [5, 13] and their references for the case of symmetric second-order elliptic operators in divergence form with smooth coe cients on bounded open sets ⊂ RN subject to Dirichlet and Neumann boundary conditions on the boundary ∂.”
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications (2016, unknown): “…has been a substantial revision of a big amount of problems in diﬀerential equations where the Laplacian is replaced by the fractional Laplacian or more general integro-diﬀerential operators, see for example [3, 5, 9, 21, 22, 23, 24, 25, 28, 29] and references therein for models and techniques.”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, background): “For A(x) being an identity matrix, the operator L becomes the well-known fractional Laplacian operator (−∆), and the detailed study about the (−∆) is available in [3, 6, 7, 8, 38, 39, 41, 42, 43].”
Regularity of spectral fractional Dirichlet and Neumann problems (2014, background): “n (such as (−∆)a and its parametrices) in HolderZygmund spaces has been known for many years, cf. e.g. Yamazaki [Y86] for a more general result and references to earlier contributions. On this point, [CS14] refers to [CS07]. It should also be possible to draw conclusions for less smooth situations t”