arXiv subjects
Publications and source records attributed to Sven Jarohs.
In this paper, we prove foliated Schwarz symmetry of solutions to a cooperatively coupled system of equations involving nonlocal operators. Here, the class of nonlocal operators covers in particular the case of the fractional Laplacian. Moreover, we give an explicit example of a nonlocal nonlinear system, in which our result can be applied.
We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.
In this work we study a class of nonlocal quadratic forms given by \[ \mathcal{E}_j(u,v)=\frac{1}{2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}(u(x)-u(y))(v(x)-v(y))j(x-y)\ dxdy, \] where $j:\mathbb{R}^N\to[0,\infty]$ is a measurable even function with $\min\{1,|\cdot|^2\}j\in L^1(\mathbb{R}^N)$. Assuming merely $j\notin L^1(\mathbb{R}^N)$, we show local compactness of the embedding $\mathcal{D}^j(\mathbb{R}^N)\hookrightarrow L^2(\mathbb{R}^N)$, where $\mathcal{D}^j(\mathbb{R}^N)$ denotes the space of functions $u\in L^2(\mathbb{R}^N)$ with $\mathcal{E}_j(u,u)<\infty$. Using this local compactness, we establish an alternative which allows to distinguish vanishing and nonvanishing of bounded sequences in $\mathcal{D}^j(\mathbb{R}^N)$. As an application, we show the existence of maximizers for a class of integral functionals defined on the unit sphere in $\mathcal{D}^j(\mathbb{R}^N)$. Our main results extend to cylindrical unbounded sets of the type $Ω= U \times \mathbb{R}^k$, where $U \subset \mathbb{R}^{N-k}$ is open and bounded. Finally, we note that a Poincaré inequality associated with $\mathcal{E}_j$ holds for unbounded domains of this type, thereby extending previously known results for bounded domains.
In this paper we derive a strong maximum principle for weak supersolutions of nonlocal equations of the form $Iu=c(x) u$ in $Ω$, where $Ω\subset \mathbb{R}^N$ is a domain, $c\in L^{\infty}(Ω)$ and $I$ is an operator of the form $Iu(x)=P.V.\int_{\mathbb{R}^N}(u(x)-u(y))j(x-y)\ dy$ with a nonnegative kernel function $j$. We formulate minimal positivity assumptions on $j$ corresponding to a class of operators which includes highly anisotropic variants of the fractional Laplacian. Somewhat surprisingly, this problem leads to the study of general lattices in $\mathbb{R}^N$. Our results extend to the regional variant of the operator $I$ and, under weak additional assumptions, also to the case of $x$-dependent kernel functions.
We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if $s$ is a natural number. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of $s$-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.
We study existence, regularity, and qualitative properties of solutions to linear problems involving higher-order fractional Laplacians $(-Δ)^s$ for any $s>1$. Using the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for $s\in(n,n+1)$ with $n\in\mathbb N$ odd; moreover, using a representation formula for solutions, we derive regularity and positivity preserving properties whenever the domain is the whole space or a ball. In the case of the whole space we analyze the Riesz kernel, which provides a fundamental solution, while in the case of the ball we show the validity of Boggio's representation formula for all integer and fractional powers of the Laplacian $s>0$. Our proofs rely on characterizations of $s$-harmonic functions using higher-order Martin kernels, on a decomposition of Boggio's formula, and on elliptic regularity theory.
We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.
In the present work we study solutions of the problem $-(-Δ)^{α/2}u = f(x,u)$ in $D_0\setminus \overline{D}_1$, with exterior conditions $u = 0$ in $R^N \setminus D_0$ and $u = 1$ in $\overline{D}_1$, where $D_1, D_0 \subset R^N$ are open sets such that $\overline{D}_1 \subset D_0$, $α\in (0,2)$, and $f$ is a nonlinearity. Under different assumptions on $f$ we prove that, if $D_0$ and $D_1$ are starshaped with respect to the same point $\bar{x} \in \overline{D}_1$, then the same occurs for every superlevel set of $u$.
In the following we show the strong comparison principle for the fractional $p$-Laplacian, i.e. we analyze functions $v,w$ which satisfy $v\geq w$ in $\mathbb{R}^N$ and \[ (-Δ)^s_pv+q(x)|v|^{p-2}v\geq (-Δ)^s_pw+q(x)|w|^{p-2}w \quad \text{in $D$,} \] where $s\in(0,1)$, $p>1$, $D\subset \mathbb{R}^N$ is an open set, and $q\in L^{\infty}(\mathbb{R}^N)$ is a nonnegative function. Under suitable conditions on $s,p$ and some regularity assumptions on $v,w$ we show that either $v\equiv w$ in $\mathbb{R}^N$ or $v>w$ in $D$. Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.
Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than other notions used so far in the literature for powers larger than 1; in particular, this evaluation can be applied to more general boundary value problems and we exhibit explicit examples. We also provide a natural variational framework and, using an asymptotic analysis, we prove how these hypersingular integrals reduce to polyharmonic operators in some cases. Our presentation aims to be as self-contained as possible and relies on elementary pointwise calculations and known identities for special functions.
For open radial sets $Ω\subset \mathbb{R}^N$, $N\geq 2$ we consider the nonlinear problem \[ (P)\quad Iu=f(|x|,u) \quad\text{in $Ω$,}\quad u\equiv 0\quad \text{on $\mathbb{R}^N\setminus Ω$ and }\lim_{|x|\to\infty} u(x)=0, \] where $I$ is a nonlocal operator and $f$ is a nonlinearity. Under mild symmetry and monotonicity assumptions on $I$, $f$ and $Ω$ we show that any continuous bounded solution of $(P)$ is axial symmetric once it satisfies a simple reflection inequality with respect to a hyperplane. In the special case where $f$ does not depend on $|x|$, we show that any nonnegative nontrivial continuous bounded solution of $(P)$ in $\mathbb{R}^N$ is radially symmetric (up to translation) and strictly decreasing in its radial direction. Our proves rely on different variants of maximum principles for antisymmetric supersolutions. As an application, we prove an axial symmetry result for minimizers of an energy functional associated to $(P)$.
We consider the nonlinear problem \[(P) \;\; I u=f(x,u) \text{ in $Ω$,} \;\; u=0 \text{ on $\mathbb{R}^{N}\setminusΩ$ }\] in an open bounded set $Ω\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on $I$, $Ω$ and the nonlinearity $f$ with respect to a fixed direction, say $x_1$, and we show that any nonnegative weak solution $u$ of $(P)$ is symmetric in $x_1$. Moreover, we have the following alternative: Either $u\equiv 0$ in $Ω$, or $u$ is strictly decreasing in $|x_1|$. The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems.
We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-Δ)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $Ω$ together with Dirichlet boundary conditions on $\R^N \setminus Ω$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $Ω$ is symmetric with respect to reflection at a hyperplane, say ${x_1=0}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak additional technical assumptions, we then show that any nonzero element in the $ω$-limit set of nonnegative globally bounded solution is even in $x_1$ and strictly decreasing in $|x_1|$. This result, which is obtained via a series of new estimates for antisymmetric supersolutions of a corresponding family of linear equations, implies a strong maximum type principle which is not available in the non-fractional case $s=1$.