This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).
In the present paper we obtain growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue in bounded non-Lipschitz domains. The proposed method is based on composition operators generated by quasiconformal mappings and their applications to weighted Sobolev inequalities.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.
We consider an optimal control problem governed by a class of boundary value problem with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes is stated. The proof of the main result relies on variational structure of the problem. To show that boundary value problem with the Dirichlet fractional Laplacian has a weak solution we employ the renowned Ky Fan Theorem.
We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $Ω\subset\mathbb R^2$. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in the frameworks of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains in terms of conformal (hyperbolic) geometry.
We construct Green functions of Dirichlet boundary value problems for sub-Laplacians on certain unbounded domains of a prototype Heisenberg-type group (prototype H-type group, in short). We also present solutions in an explicit form of the Dirichlet problem for the sub-Laplacian with non-zero boundary datum on half-space, quadrant-space, etc, as well as in a strip.
Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.
In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by Payne-Pólya-Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh-Benguria.
In this paper we analyze the asymptotic behavior of the Dirichlet fractional Laplacian $(-Δ_{\mathbb R^{n+k}})^{s}$, with $s\in (0, 1)$, on bounded domains in $\mathbb R^{n+k}$ that become unbounded in the last $k$-directions. A dimension reduction phenomenon is observed and described via $Γ$-convergence.
In the whole space $\mathbb R^d$, linear estimates for heat semi-group in Besov spaces are well established, which are estimates of $L^p$-$L^q$ type, maximal regularity, e.t.c. This paper is concerned with such estimates for semi-group generated by the Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of $\mathbb R^d$.
We investigate a Dirichlet problem for the Laplace equation in a domain of $\mathbb{R}^2$ with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance $|ε_1|$ one from the other and each one of size $|ε_1ε_2|$. In such a domain, we introduce a Dirichlet problem and we denote by $u_{ε_1,ε_2}$ its solution. We show that the dependence of $u_{ε_1,ε_2}$ upon $(ε_1,ε_2)$ can be described in terms of real analytic maps of the pair $(ε_1,ε_2)$ defined in an open neighborhood of $(0,0)$ and of logarithmic functions of $ε_1$ and $ε_2$. Then we study th…
In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the degree theory after turning the problem into a Hammerstein equation. The right hand side $g$ is a possibly discontinuous function in the second variable satisfying some non-standard growth conditions..
In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fractional operators with different orders include medicine, plasma physics, and population dynamics.
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.
We obtain a criterion for the existence of solutions of the problem $$ Δ_p u = 0 \quad \mbox{in } M \setminus \partial M, \quad \left. u \right|_{ \partial M } = h, $$ with the bounded Dirichlet integral, where $M$ is an oriented complete Riemannian manifold with boundary and $h \in W_{p, loc}^1 (M)$, $p > 1$.
In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-Δ)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with symbol $2\log|ξ|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-nor…
In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we analyzed the $Γ$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\rightarrow+\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $Γ$-converges to a minimizer of Hölder $\infty$-Laplacian under the same Dirichlet boundary condition. On the other hand, we first investigate the asymptotic behaviour of non-homogeneous fractional $p$-fun…
We consider an optimal control problem of diffusion equation with missing data governed by the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary interaction domain disjoint from the domain of the state equation. We assume that the unknown initial condition belongs to an appropriate space of infinite dimension, the so-called space of uncertainties. The key tools we used in order to characterize the optimal control is the no-regret and low-regret control developed by J.L Lions.
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.
The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order problems are less rigid and more complex. One way to better understand this transition is to study the intermediate Dirichlet problem in terms of fractional Laplacians. This survey aims to be an introduction to these type of problems; in particular, the different pointwise notions for these operators is introduced considering a suitable natural extension of …
We consider the principal eigenvalue problem for the Laplace-Beltrami operator on the upper half of a topological torus under the Dirichlet boundary condition. We present a construction of the upper half of a topological torus that admits the principal eigenfunction having exact numbers of critical points. Furthermore, we manage to identify the locations of all the critical points of the principal eigenfunction explicitly.
On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Brandenburg University of Technology Cottbus–Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格分校）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive solutions as the parameter $λ>0$ varies, is proven. Since for every admissible $λ$ the problem has a smallest positive solution $\bar u_λ$, both monotonicity and continuity of the map $ λ\mapsto \bar u_λ$ are studied.
A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such that, for every $Ω\subset\mathbb{R}^n$ open, bounded and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds \begin{equation*} λ_1(Ω)-λ_1(B)\geq c\left(P(Ω)-P(B) \right)^{2}, \end{equation*} where by $λ_1(\cdot)$ we denote the first Dirichlet eigenvalue of a set and by $P(\cdot)$ its perimeter. The hearth of the present pap…
In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit Neumann problem. The first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. Then we focus on the case of Diri…
In this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian $λ$ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane…
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature…
We provide bounds for the sequence of eigenvalues $\{λ_i(Ω)\}_i$ of the Dirichlet problem $$ L_Δu=λu\ \ {\rm in}\ \, Ω,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^N\setminus Ω,$$ where $L_Δ$ is the logarithmic Laplacian operator with Fourier transform symbol $2\ln |ζ|$. The logarithmic Laplacian operator is not positively definitive if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first $k$ eigenvalues by extending the Li-Yau method and Kröger's method respectively. Moreover, we show the limit of the sum of the first $k$ eigenva…
We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-Δ)^s$ on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\textrm{loc}}(Ω)$, while for $2\leq p<\infty$ we show that the solutions belong to $W^{2s,p}_{\textrm{loc}}(Ω)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working dir…
In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^α, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om. \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > sp, s \in (0,1), \la >0, 0 < q \leq 1 $ and $α\le p^*_s-1$. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter $\la$.
In this paper, we study the logarithmic Laplacian operator $L_Δ$, which is a singular integral operator with symbol $2\log |ζ|$. We show that this operator has the integral representation $$L_Δu(x) = c_{N} \int_{\mathbb{R}^N } \frac{ u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^{N} } dy + ρ_N u(x) $$ with $c_N = π^{- \frac{N}{2}} Γ(\frac{N}{2})$ and $ρ_N=2 \log 2 + ψ(\frac{N}{2}) -γ$, where $Γ$ is the Gamma function, $ψ= \frac{Γ'}Γ$ is the Digamma function and $γ= -Γ'(1)$ is the Euler Mascheroni constant. This operator arises as formal derivative $\partial_s \Big|_{s=0} (-Δ)^s$ of fractional Laplacians at $s…
We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -Δ_1 u =h(u)f(x)&\hbox{in }Ω\,,\\ u=0&\hbox{on }\partialΩ\,, \end{array}\right. $$ where $ Ω\subset \mathbb{R}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(Ω)$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions.
We consider the Dirichlet-energy maximization problem of the solution $u_f$ of (\ref{main}), among all functions $f\in L^2(D)$, such that $\|f\|_2= 1$. We show that the two maximizers are the first eigenfunctions of the Laplacian with Dirichlet boundary condition $f=\pm u_1$.
For $s>-1$, $s\notin\mathbb N_0$, we compare two natural types of fractional Laplacians $(-Δ)^s$, namely, the restricted Dirichlet and the spectral Neumann ones. We show that for the quadratic form of their difference taken on the space $\tilde{H}^s(Ω)$ is positive or negative depending on whether the integer part of $s$ is even or odd. For $s\in(0,1)$ and convex domains we prove also that the difference of these operators is positivity preserving on $\tilde{H}^s(Ω)$. This paper complements [10] and [11] where similar statements were proved for the spectral Dirichlet and the restricted Dirichl…
