Gerd Grubb

We prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ( $$0<a<1$$ 0 < a < 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-\Delta)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $\Omega$ in $R^n$. The presentation starts at an elementary level. Two points are explained in detail: 1) How the factor $d^a$, with $d(x)=dist(x,d\Omega)$, comes into the picture, related to the fact that the precise solution spaces for the homogeneous Dirichlet problem are so-called a-transmission spaces. 2) The natural definition of a local nonhomogeneous Dirichlet condition $\gamma_0(u/d^{a-1})=\varphi$. We also give brief accounts of some further developments: Evolution problems (for $d_t u - r^+Pu = f(x,t)$) and resolvent problems (for $Pu-\lambda u=f$), also with nonzero boundary conditions. Integration by parts, Green's formula.

The fractional Laplacian $(-\Delta )^a$, $a\in(0,1)$, and its generalizations to variable-coefficient $2a$-order pseudodifferential operators $P$, are studied in $L_q$-Sobolev spaces of Bessel-potential type $H^s_q$. For a bounded open set $\Omega \subset \mathbb R^n$, consider the homogeneous Dirichlet problem: $Pu =f$ in $\Omega $, $u=0$ in $ \mathbb R^n\setminus\Omega $. We find the regularity of solutions and determine the exact Dirichlet domain $D_{a,s,q}$ (the space of solutions $u$ with $f\in H_q^s(\overline\Omega )$) in cases where $\Omega $ has limited smoothness $C^{1+\tau }$, for $2a<\tau <\infty $, $0\le s<\tau -2a$. Earlier, the regularity and Dirichlet domains were determined for smooth $\Omega$ by the second author, and the regularity was found in low-order H\"older spaces for $\tau =1$ by Ros-Oton and Serra. The $H_q^s$-results obtained now when $\tau <\infty $ are new, even for $(-\Delta )^a$. In detail, the spaces $D_{a,s,q}$ are identified as $a$-transmission spaces $H_q^{a(s+2a)}(\overline\Omega )$, exhibiting estimates in terms of $\operatorname{dist}(x,\partial\Omega )^a$ near the boundary. The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.

For $2a$-order strongly elliptic operators $P$ generalizing $(-\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\Omega \subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<\infty $, when $\Omega $ is $C^{\tau +1}$ with a finite $\tau >2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar results for $P-\lambda I$, $\lambda \in C$. The solution spaces equal $a$-transmission spaces $H_q^{a(s+2a)}(\bar\Omega)$. Similar results are shown for nonhomogeneous Dirichlet problems, prescribing the local Dirichlet trace $(u/d^{a-1})|_{\partial\Omega }$, $d(x)=dist(x,\partial\Omega)$. They are solvable in the larger spaces $H_q^{(a-1)(s+2a)}(\bar\Omega)$. Moreover, the nonhomogeneous problem with a spectral parameter $\lambda \in C$, $$ Pu-\lambda u = f \text { in }\Omega ,\quad u=0 \text { in }R^n\setminus \Omega ,\quad (u/d^{a-1 })|_{\partial\Omega }=\varphi \text{ on }\partial\Omega , $$ is for $q<(1-a)^{-1}$ shown to be uniquely resp. Fredholm solvable when $\lambda $ is in the resolvent set resp. the spectrum of the $L_2$-Dirichlet realization. Finally, we show solvability results for evolution problems $Pu+d_tu= f(x,t)$ in $L_2$ and $L_q$-based spaces over $C^{1+\tau}$-domains, including nonhomogeneous local boundary conditions.

The paper treats pseudodifferential operators $ P = \operatorname{Op}(p(\xi)) $ with homogeneous complex symbol $ p(\xi) $ of order $ 2a > 0 $, generalizing the fractional Laplacian $ (-\Delta)^a $ but lacking its symmetries, and taken to act on the halfspace ${\mathbb R}^n_+$. The operators are seen to satisfy a principal $ \mu $-transmission condition relative to ${\mathbb R}^n_+$, but generally not the full $ \mu $-transmission condition satisfied by $ (-\Delta)^a $ and related operators (with $ \mu = a $). However, $ P $ acts well on the so-called $ \mu $-transmission spaces over ${\mathbb R}^n_+$ (defined in earlier works), and when $ P $ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $ P $, leading to regularity results with a factor $ x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over ${\mathbb R}^n_+$ for $ P $ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal $ \mu $-transmission condition has weaker requirements than the full $ \mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.

The paper treats pseudodifferential operators $P=Op(p(\xi ))$ with homogeneous complex symbol $p(\xi )$ of order $2a>0$, generalizing the fractional Laplacian $(-\Delta )^a$ but lacking its symmetries, and taken to act on the halfspace $R^n_+$. The operators are seen to satisfy a principal $\mu $-transmission condition relative to $R^n_+$, but generally not the full $\mu $-transmission condition satisfied by $(-\Delta )^a$ and related operators (with $\mu =a$). However, $P$ acts well on the so-called $\mu $-transmission spaces over $R^n_+$ (defined in earlier works), and when $P$ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $P$, leading to regularity results with a factor $x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over $R^n_+$ for $P$ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, for this new class of operators. Since the principal $\mu $-transmission condition has weaker requirements than the full $\mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.

For a strongly elliptic pseudodifferential operator $L$ of order $2a$ ($0<a0$. Here we deduce a "halfways Green's formula" for $L$: $$ \int_{R^n_+} Lu\,\bar v\,dx-\int_{R^n_+}u\,\overline{ L^*v}\,dx=c\int_{R^{n-1}}\gamma_0(u/x_n^{\mu -1 })\,{\gamma_0(\bar v/x_n^{\mu ^*})}\, dx', $$ when $u$ solves a nonhomogeneous Dirichlet problem for $L$, and $v$ solves a homogeneous Dirichlet problem for $L^*$; $\mu ^*=2a-\mu $. Finally, we show a full Green's formula, when both $u$ and $v$ solve nonhomogeneous Dirichlet problems; here both Dirichlet and Neumann traces of $u$ and $v$ enter, as well as a first-order pseudodifferential operator over the boundary.

When $ P$ is the fractional Laplacian $ (-\Delta )^a$, $ 0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $ \Omega \subset{\Bbb R}^n$: $ r^+Pu(x, t)+\partial_tu(x, t) = f(x, t)$ on $ \Omega \times \, ]0, T[\, $, $ u(x, t) = 0$ for $ x\notin\Omega$, $ u(x, 0) = 0$, is known to be solvable in relatively low-order Sobolev or Hölder spaces. We now show that in contrast with differential operator cases, the regularity of $ u$ in $ x$ at $ \partial\Omega$ when $ f$ is very smooth cannot in general be improved beyond a certain estimate. An improvement requires the vanishing of a Neumann boundary value. --- There is a similar result for the Schrödinger Dirichlet problem $ r^+Pv(x)+Vv(x) = g(x)$ on $ \Omega$, $ \text{supp } v\subset\overline\Omega$, with $ V(x)\in C^\infty$. The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a $ \text{dist}(x, \partial\Omega )^a$ singularity.