Propagation of Singularities for Generalized Solutions to Wave Equations with Discontinuous Coefficients

Viscosity Solutions of Hamilton–jacobi Equations with Discontinuous Coefficients

Abstract. We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to “interna...

Propagation, Reflection, and Diffraction of Singularities of Solutions to Wave Equations

Fundamental Solutions to Some Elliptic Equations with Discontinuous Senior Coefficients and an Inequality for These Solutions

Let Lu := ∇ · (a(x)∇u) = −δ(x − y) in R3 , 0 < c1 a(x) c2 , a(x) is a piecewise-smooth function with the discontinuity surface S which is smooth. It is proved that in an neighborhood of S the behavior of the function u is given by the formula: u(x, y) = ⎧⎨ ⎩ (4πa+)−1[r−1 xy + bR−1], y3 > 0, (4πa−)−1[r−1 xy − bR−1], y3 < 0. ( ∗ ) Here the local coordinate system is chosen in which the origin lie...

Stability for Solutions of Wave Equations with C1,1 Coefficients

We consider the stable dependence of solutions to wave equations on metrics in C class. The main result states that solutions depend uniformly continuously on the metric, when the Cauchy data is given in a range of Sobolev spaces. The proof is constructive and uses the wave packet approach to hyperbolic equations.

Gradient Estimates for Solutions to Divergence Form Elliptic Equations with Discontinuous Coefficients

In this paper we derive globalW 1,∞ and piecewiseC1,α estimates for solutions to divergence form elliptic equations with piecewise Hölder continuous coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.