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Register a new userSymmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region
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We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and post-process the basis, rendering it suitable for input to the GNGA. The GNGA uses Newtons method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE and the 59 generic symmetry-breaking bifurcations among these symmetry types. Our numerical code uses continuation methods, and follows branches created at symmetry-breaking bifurcations, so the human user does not need to supply initial guesses for Newtons method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.
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We seek solutions $uinR^n$ to the semilinear elliptic partial difference equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) {it Nonlinear Elliptic Partial Difference Equations on Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) {it Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region} wherein we present some of our recent advances concerning symmetry, bifurcation, and automation fo We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newtons method, create informative graphics, and be 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-dimension we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithms capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelis of a graph. We highlight interesting symmetry and variational phenomena.
In this paper we numerically solve the eigenvalue problem $Delta u + lambda u = 0$ on the fractal region defined by the Koch Snowflake, with zero-Dirichlet or zero-Neumann boundary conditions. The Laplacian with boundary conditions is approximated by a large symmetric matrix. The eigenvalues and eigenvectors of this matrix are computed by ARPACK. We impose the boundary conditions in a way that gives improved accuracy over the previous computations of Lapidus, Neuberger, Renka & Griffith. We extrapolate the results for grid spacing $h$ to the limit $h rightarrow 0$ in order to estimate eigenvalues of the Laplacian and compare our results to those of Lapdus et al. We analyze the symmetry of the region to explain the multiplicity-two eigenvalues, and present a canonical choice of the two eigenfunctions that span each two-dimensional eigenspace.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
We revisit some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case we give a new proof of the uniqueness result assuming only a positive lower bound for Ricci curvature. We apply the same principle in the CR case and reconstruct the Jerison-Lee identity in a more general setting. As a consequence we prove a stronger uniqueness result in the CR case. We also discuss some open problems for further study.
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is ill-posed, in the sense that solutions do not typically exist forward or backward in time. In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which applies to infinite cylindrical (channel) domains, and also generalizes previous work on radial domains as we impose no symmetry assumptions on the equation or its solutions.
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