arXiv subjects
Publications and source records attributed to Tonia Ricciardi.
We use a shadowing-type lemma in order to analyze the singular, semilinear elliptic equation describing static self-dual abelian Higgs vortices. Such an approach allows us to construct new solutions having an \textit{infinite} number of arbitrarily prescribed vortex points. Furthermore, we obtain the precise asymptotic profile of the solutions in the form of an approximate superposition rule, up to an error which is exponentially small.
We prove a sharp Hölder estimate for solutions of linear two-dimensional, divergence form elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has {\em unit determinant}. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.
We prove the existence of at least two doubly periodic vortex solutions for a self-dual CP(1) Maxwell-Chern-Simons model. To this end we analyze a system of two elliptic equations with exponential nonlinearities. Such a system is shown to be equivalent to a fourth-order elliptic equation admitting a variational structure.
We prove the existence of at least two solutions for a fourth order equation, which includes the vortex equations for the U(1) and CP(1) self-dual Maxwell-Chern-Simons models as special cases. Our method is variational, and it relies on an "asymptotic maximum principle" property for a special class of supersolutions to this fourth order equation.
We define an abstract nonlinear elliptic system, admitting a variational structure, and including the vortex equations for some Maxwell-Chern-Simons gauge theories as special cases. We analyze the asymptotic behavior of its solutions, and we provide a general simplified framework for the asymptotics previously derived in those special cases. As a byproduct of our abstract formulation, we also find some new qualitative properties of solutions.
Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} \|u\|_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in H^1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], R_g is the scalar curvature, $K^{-1}=\inf\|\nabla u\|_{L^2(\R^n)}\|u\|_{L^{2n/(n-2)}(\R^n)}^{-1}$ and A>0 is a constant depending on (M,g) only. The inequality is {\em sharp} in the sense that on any (M,g), $K$ can not be replaced by any smaller number and R_g can not be replaced by any continuous function which is smaller than R_g at some point. If (M,g) is not locally conformally flat, the exponent 2n/(n+2) can not be replaced by any smaller number. If (M,g) is locally conformally flat, a stronger inequality, with 2n/(n+2) replaced by 1, holds in all dimensions n>=3.