Fully non linear equations with integral diffusions
نویسندگان
چکیده
The authors establish weighted decay rates for the energy associated with the Cauchy problem { utt − div(b(x)∇u) + a(x)ut = 0, x ∈ R, t > 0 u(0, x) = u0(x), ut(0, x) = u1(x), x ∈ R. Such a system appears in models for traveling waves in a non-homogeneous gas with damping that changes with the position. This problem has been studied intensively for the homogeneous medium (when b is constant), but the results are scarce for the variable coefficient case. In fact, to the authors’ knowledge, the results of this paper are the first to be obtained for wave equations with variable coefficients on the entire space R when damping is present. The proof is based on the multiplier method where the multiplier is cleverly engineered from an asymptotic profile of solutions. These results were obtained in collaboration with G. Todorova and B. Yordanov from University of Nebraska-Lincoln. Raynor, Sarah Wake Forest University Title: Neumann fixed boundary regularity for an elliptic free boundary problem Abstract: We examine the regularity properties of solutions to an elliptic free boundary problem near a Neumann fixed boundary. Consider a nonnegative function u, defined variationally, which is harmonic where it is positive and satisfies a gradient jump condition weakly along the free boundary ∂{u > 0}. Our main result is that u is Lipschitz continuous. Additionally, we prove various basic properties of such a minimizer near a portion of the fixed boundary on which ∂u ∂ν = 0 weakly. Our results include up-to-the boundary gradient estimates on harmonic functions with Neumann boundary conditions on convex domains, which have independent interest. Shen, Chun-Yen Indiana University Title: An explicit sum product estimate in Fp Abstract: Let A be a subset of Fp, the field of p elements with p prime. We let A+ A = {a+ b : a ∈ A, b ∈ A}, and AA = {ab : a ∈ A, b ∈ A}. Bourgain, Katz and Tao have shown that if |A| < p1−δ, where δ > 0, then one has the sum product estimate max(|A+ A|, |AA|) ≥ |A|
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