The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain AllenCahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the ...

We study the properties of the "rigid Laplacian" operator; that is we consider solutions of the Laplacian equation in the presence of fixed truncation errors. The dynamics of convergence to the correct analytical solution displays the presence of a metastable set of numerical solutions, whose presence can be related to granularity. We provide some scaling analysis in order to determine the valu...

We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem −∆pu = au p−1 − b(x)u, u|∂Ω = 0 as q → p − 1 + 0 and as q → ∞ via a scale argument. Here ∆p is the p-Laplacian with 1 < p < ∞ and q > p−1. If p = 2, such problems arise in population dynamics. Our main results generalize the results for p = 2, but some technical difficulties arising from the nonlinear d...

This paper treats the antimaximum principle and the existence of a solution for quasilinear elliptic equation −div (a(x, |∇u|)∇u) = λm(x)|u|p−2u+ h(x) in Ω under the Neumann boundary condition. Here, a map a(x, |y|)y on Ω×RN is strictly monotone in the second variable and satisfies certain regularity conditions. This equation contains the p -Laplacian problem as a special case. Mathematics subj...

The paper concerns the Hamiltonian structure of the finite-dimensional reductions 2D dispersionless Toda hierarchy constrained by the string equation. We derive the Hamiltonian structure of the reduced dynamics and show connections of integrals of “multi-finger” solutions of the Laplacian growth problem with the “Toda–Krichever” flows of the 2dToda hierarchy constrained by a string equation. Th...

We construct a wavelet basis on the unit interval with respect to which both the (infinite) mass and stiffness matrix corresponding to the one-dimensional Laplacian are (truly) sparse and boundedly invertible. As a consequence, the (infinite) stiffness matrix corresponding to the Laplacian on the n-dimensional unit box with respect to the n-fold tensor product wavelet basis is also sparse and b...

Experiments in quasi-two-dimensional geometry (Hele-Shaw cells) in which a fluid is injected into a viscoelastic medium (foam, clay, or associating polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a prerequisite to describing the fracture of elastic material is the solu...

The Seiberg-Witten equations are studied from the viewpoint of gauge potential decomposition. We find a determinant equation ∆Aμ = −λAμ for the twisting U(1) potential Aμ of the Seiberg-Witten theory, which is in itself an eigenvalue problem of the Laplacian operator, with the eigenvalue being the vacuum expectation value of the field function, λ = ‖Φ‖ /2. This establishes a direct relationship...

We prove that for the Martinet wave equation with “flat” metric, which is a subelliptic equation, singularities can propagate at any speed between 0 and 1 along singular geodesic. This in strong contrast usual propagation of equations elliptic Laplacian.

its theory is well known, among its features we find C∞ smoothness of solutions, infinite speed of propagation of disturbances and the strong maximum principle. These properties are able to be generalized to a number of related evolution equations, notably those which are linear and uniformly parabolic. Other wellknown examples of (1) include the porous medium equation ut = △u, m > 1, (4) and e...