Semilinear Elliptic PDE Research Articles

Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs

Variational gluing arguments are employed to construct new families of solutions for a class of semilinear elliptic PDEs. The main tools are the use of invariant regions for an associated heat flow and variational arguments. The latter provide a characterization of critical values of an associated functional. Among the novelties of the paper are the construction of “hybrid” solutions by gluing minima and mountain pass solutions and an analysis of the asymptotics of the gluing process.

Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

We seek discrete approximations to solutions $u:\Omega\to\mathbb{R}$ of semilinear elliptic PDE of the form $\Delta u+f_s(u)=0$, where $f_s$ is a one-parameter family of nonlinear functions and $\Omega$ is a domain in $\mathbb{R}^d$. The main achievement of this paper is the approximation of solutions to the PDE on the cube $\Omega=(0,\pi)^3\subseteq\mathbb{R}^3$. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531--2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton--Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801--820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443--464].

Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs

AbstractWe investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor‐, Legendre‐ and Chebyshev polynomials on the infinite‐dimensional parameter domain. We deduce rates of convergence for N term truncated approximations of expansions of the parametric solution. We also deduce spatial regularity of the solution, and establish convergence rates of N‐term discretizations of the parametric solutions with respect to these polynomials in parameter space and with respect to a multilevel hierarchy of finite element spaces in the spatial domain of the PDE.

Smooth bifurcation for a Signorini problem on a rectangle

We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof are first a certain local equivalence of the unilateral BVP to a system consisting of a corresponding classical BVP and of two scalar equations (which determine the ends of the contact intervals), and secondly an application of the classical Crandall-Rabinowitz type local bifurcation techniques (scaling and application of the Implicit Function Theorem) to that system.

HOMOGENIZATION OF A PERIODIC DEGENERATE SEMILINEAR ELLIPTIC PDE

In this paper, a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of ℝd. Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.

A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.

A note on hybrid heteroclinic solutions for a class of semilinear elliptic PDEs

Variational methods will be used to find solutions of a semilinear elliptic PDE by ‘‘glueing’’ mountain pass critical points of the corresponding functional to minima or to other mountain pass critical points.

A semilinear elliptic PDE not in divergence form via variational methods

In this paper we consider a semilinear equation driven by an operator not in divergence form. Precisely, the principal part of the operator is in divergence form, but it has also a lower order term depending on Du. While the right-hand side of the equation satisfies superlinear and subcritical growth conditions at zero and at infinity. The problem has not a variational structure, but, despite that, we use variational techniques in order to prove an existence and regularity result for the equation.

Single and Multitransition Solutions for a Family of Semilinear Elliptic PDE’s

These lectures survey a class of partial differential equations whose study was initiated by Moser. This class includes various Allen-Cahn models of phase transitions. In line with this connection, the existence of a large variety of solutions which undergo spatial transitions is established. The main tools are variational arguments, especially minimization methods.

Radial symmetry of solutions for some integral systems of Wolff type

We consider the fully nonlinear integral systems involving Wolff potentials: $\u(x) = W_{\beta, \gamma}(v^q)(x)$, $\x \in R^n$; $\v(x) = W_{\beta, \gamma} (u^p)(x)$, $\x \in R^n$; (1) where $ \W_{\beta,\gamma} (f)(x) = \int_0^{\infty}$ $[ \frac{\int_{B_t(x)} f(y) dy}{t^{n-\beta\gamma}} ]^{\frac{1}{\gamma-1}} \frac{d t}{t}.$ After modifying and refining our techniques on the method of moving planes in integral forms, we obtain radial symmetry and monotonicity for the positive solutions to systems (1). This system includes many known systems as special cases, in particular, when $\beta = \frac{\alpha}{2}$ and $\gamma = 2$, system (1) reduces to $\u(x) = \int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}} v(y)^q dy$, $ x \in R^n$, $v(x) = \int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}} u(y)^p dy$, $ x \in R^n$. (2) The solutions $(u,v)$ of (2) are critical points of the functional associated with the well-known Hardy-Littlewood-Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs $(-\Delta)^{\alpha/2} u = v^q$, in $R^n$, $(-\Delta)^{\alpha/2} v = u^p$, in $R^n$ (3) which comprises the well-known Lane-Emden system and Yamabe equation.

Regularity of solutions for an integral system of Wolff type

We consider the fully nonlinear integral systems involving Wolff potentials: (1) { u ( x ) = W β , γ ( v q ) ( x ) , x ∈ R n , v ( x ) = W β , γ ( u p ) ( x ) , x ∈ R n ; where W β , γ ( f ) ( x ) = ∫ 0 ∞ [ ∫ B t ( x ) f ( y ) d y t n − β γ ] 1 γ − 1 d t t . This system includes many known systems as special cases, in particular, when β = α 2 and γ = 2 , system (1) reduces to (2) { u ( x ) = ∫ R n 1 | x − y | n − α v ( y ) q d y , x ∈ R n , v ( x ) = ∫ R n 1 | x − y | n − α u ( y ) p d y , x ∈ R n . The solutions ( u , v ) of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs { ( − Δ ) α / 2 u = v q , in R n , ( − Δ ) α / 2 v = u p , in R n , which comprises the well-known Lane–Emden system and Yamabe equation. We obtain integrability and regularity for the positive solutions to systems (1) . A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.

On Green’s functions for positive, elliptic differential operators on closed, Riemannian manifolds and some consequences for semi-linear PDE

In this article, we establish the non-negativity of Green’s functions for a class of elliptic differential operators on closed, Riemannian manifolds. The method used is the calculus of variations with a unilateral constraint. Consequences of this result are then explored, with a maximum principle-type result established for a broad class of semi-linear elliptic PDE as a result. Finally, a result regarding the positivity of the principal eigenfunction of an operator in this class is proven as well.

AUTOMATED BIFURCATION ANALYSIS FOR NONLINEAR ELLIPTIC PARTIAL DIFFERENCE EQUATIONS ON GRAPHS

We seek solutions u ∈ ℝn to the semilinear elliptic partial difference equation -Lu + fs(u) = 0, where L is the matrix corresponding to the Laplacian operator on a graph G and fs is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: (a) Nonlinear elliptic partial difference equations on graphs (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and (b) Symmetry and automated branch following for a semilinear elliptic PDE on a fractal region (SIAM J. Dynamical Systems, 2006), wherein we present some of our recent advances concerning symmetry, bifurcation and automation for PDE. We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newton's method, create informative graphics, and better understand the role of symmetry in the underlying variational structure. We use two modified implementations of the gradient Newton–Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimensional critical eigenspaces, we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithm's capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelist of a graph. We highlight interesting symmetry and variational phenomena.

OPTIMAL HARVESTING OF A SPATIALLY EXPLICIT FISHERY MODEL

Abstract We consider an optimal fishery harvesting problem using a spatially explicit model with a semilinear elliptic PDE, Dirichlet boundary conditions, and logistic population growth. We consider two objective functionals: maximizing the yield and minimizing the cost or the variation in the fishing effort (control). Existence, necessary conditions, and uniqueness for the optimal harvesting control for both cases are established. Results for maximizing the yield with Neumann (no‐flux) boundary conditions are also given. The optimal control when minimizing the variation is characterized by a variational inequality instead of the usual algebraic characterization, which involves the solutions of an optimality system of nonlinear elliptic partial differential equations. Numerical examples are given to illustrate the results.

Flattening Results for Elliptic PDEs in Unbounded Domains with Applications to Overdetermined Problems

Semilinear elliptic PDEs are dealt with, either in the half-space or in overdetermined settings. We obtain several new results and also give new proofs of celebrated theorems by exploiting some geometric analysis of level sets, a geometric inequality and a pointwise gradient estimate.

Sufficient optimality conditions for multidimensional optimal control problems with mixed constraints governed by an elliptic PDE

In this article, the idea of a dual dynamic programming is applied to the optimal control problems with multiple integrals governed by a semi-linear elliptic PDE and mixed state-control constraints. The main result called a verification theorem provides the new sufficient conditions for optimality in terms of a solution to the dual equation of a multidimensional dynamic programming. The optimality conditions are also obtained by using the concept of an optimal dual feedback control. Besides seeking the exact minimizers of problems considered some kind of an approximation is given and the sufficient conditions for an approximated optimal pair are derived.

On the multiplicity of periodic solutions of mountain pass type for a class of semilinear PDE’s

We prove the existence of many mountain pass periodic solutions for a semilinear elliptic PDE on a torus.

On homotopy continuation method for computing multiple solutions to the Henon equation

AbstractMotivated by numerical examples in solving semilinear elliptic PDEs for multiple solutions, some properties of Newton homotopy continuation method, such as its continuation on symmetries, the Morse index, and certain functional structures, are established. Those results provide useful information on selecting initial points for the method to find desired solutions. As an application, a bifurcation diagram, showing the symmetry/peak breaking phenomena of the Henon equation, is constructed. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Oscillation Theorems for Quasilinear Elliptic Differential Equations

By using vector Riccati transformation and averaging technique, some oscillation criteria for the quasilinear elliptic di.erential equation of second order, $$ \sum\limits_{i,j = 1}^N {D_i \left[ {\Psi \left( y \right)A_{ij} \left( x \right)\left| {Dy} \right|^{p - 2} D_j y} \right] + p\left( x \right)f\left( y \right)} = 0, $$ are obtained. These theorems extend and include earlier results for the semilinear elliptic equation and PDE with p-Laplacian.

Smooth bifurcation for variational inequalities based on Lagrange multipliers

We prove a bifurcation theorem of Crandall-Rabinowitz type (local bifurcation of smooth families of nontrivial solutions) for general variational inequalities on possibly nonconvex sets with infinite-dimensional bifurcation parameter. The result is based on local equivalence of the inequality to a smooth equation with Lagrange multipliers, on scaling techniques and on an application of the implicit function theorem. As an example, we consider a semilinear elliptic PDE with nonconvex unilateral integral conditions on the boundary of the domain.