scholarly journals Semilinear elliptic equations involving mixed local and nonlocal operators

2020 ◽  
pp. 1-31
Author(s):  
Stefano Biagi ◽  
Eugenio Vecchi ◽  
Serena Dipierro ◽  
Enrico Valdinoci
Keyword(s):  
Elliptic Equations ◽  
Radial Symmetry ◽  
Order Differential Operator ◽  
Nonlocal Operator ◽  
Nonlocal Operators ◽  
One Dimensional ◽  
Symmetry Properties ◽  
Moving Plane ◽  
Symmetry Result ◽  
Fractional Type

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ) s , with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

scholarly journals Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

2021 ◽  
Vol 4 (5) ◽  
pp. 1-24
Author(s):  
Filippo Gazzola ◽  
◽  
Gianmarco Sperone ◽  
Keyword(s):  
Unit Ball ◽  
Elliptic Equations ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Dirichlet Boundary Conditions ◽  
Polyharmonic Equations ◽  
Alternative Formula ◽  
Symmetry Properties ◽  
Derivatives Of

<abstract><p>Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in <italic>conformal dimensions</italic>, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.</p></abstract>

scholarly journals On the moving plane method for boundary blow-up solutions to semilinear elliptic equations

2018 ◽  
Vol 9 (1) ◽  
pp. 1-6
Author(s):  
Annamaria Canino ◽  
Berardino Sciunzi ◽  
Alessandro Trombetta
Keyword(s):  
Weak Solutions ◽  
Elliptic Equations ◽  
Blow Up ◽  
Radial Symmetry ◽  
Semilinear Elliptic Equations ◽  
Moving Plane Method ◽  
Plane Method ◽  
Semilinear Elliptic ◽  
Annular Domains ◽  
Moving Plane

Abstract We consider weak solutions to {-\Delta u=f(u)} on {\Omega_{1}\setminus\Omega_{0}} , with {u=c\geq 0} in {\partial\Omega_{1}} and {u=+\infty} on {\partial\Omega_{0}} , and we prove monotonicity properties of the solutions via the moving plane method. We also prove the radial symmetry of the solutions in the case of annular domains.

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals Radial symmetry for an elliptic PDE with a free boundary

2021 ◽  
Vol 8 (26) ◽  
pp. 311-319
Author(s):  
Layan El Hajj ◽  
Henrik Shahgholian
Keyword(s):  
Free Boundary ◽  
Radial Symmetry ◽  
Regularity Of Solutions ◽  
Elliptic Pde ◽  
Linear Elliptic Equation ◽  
Boundary Harnack Principle ◽  
Partial Differential ◽  
Content Type ◽  
Moving Plane ◽  
Plane Technique

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u )  in  B 1 , 0 ≤ u > M ,  in  B 1 , u = M ,  on  ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

Bifurcations, symmetries and the notion of fixed subspace

2021 ◽  
pp. 2130006
Author(s):  
Giampaolo Cicogna
Keyword(s):  
Topological Index ◽  
Ad Hoc ◽  
Index Theory ◽  
Typical Case ◽  
One Dimensional ◽  
Dimension Formula ◽  
Symmetry Properties ◽  
Equivariant Bifurcation ◽  
Stationary Bifurcation ◽  
Bifurcation Problems

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

2020 ◽  
Author(s):  
Guangze Gu ◽  
Changfeng Gui ◽  
Yeyao Hu ◽  
Qinfeng Li
Keyword(s):  
Field Equation ◽  
Level Sets ◽  
Mean Field ◽  
Flat Torus ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Flat Tori ◽  
The Mean ◽  
Symmetry Result ◽  
Mean Field Equation

Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau&gt;0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

Intervalley kinetic energy terms in the effective mass equation: a one-dimensional analytical study

1989 ◽  
Vol 67 (9) ◽  
pp. 896-903 ◽  
Author(s):  
Lorenzo Resca
Keyword(s):  
Kinetic Energy ◽  
Effective Mass ◽  
Analytical Study ◽  
Three Dimensional ◽  
Real Space ◽  
The Other ◽  
Alternative Formulation ◽  
One Dimensional ◽  
Symmetry Properties ◽  
Band Case

We show that a one-dimensional analytical study allows us to test and clarify the derivation, assumptions, and symmetry properties of the intervalley effective mass equation (IVEME). In particular, we show that the IVEME is consistent with a two-band case, and is in fact exact for a model that satisfies exactly all its assumptions. On the other hand, an alternative formulation in k-space that includes intervalley kinetic energy terms is consistent with a one-band case, provided that intra-valley kinetic energy terms are also calculated consistent with one band. We also show that the standard symmetry assumptions for both real space and k-space formulations are not actually exact, but are consistent with a "total symmetric" projection, or with taking spherical averages in a three-dimensional case.

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