Existence and uniqueness of solution of free boundary problem with partially degenerate diffusion

In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.     

A variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary

We develop a variational theory to study the free boundary regularity problem for elliptic operators: Lu=Dj(a_ij(x)Diu)+biui+c(x)u=0$Lu=D_j(a_{ij}(x)D_{i}u)+b_i{u}_i+c(x)u=0$ in {u>0}$\{u>0\}$, 〈a_ij(x)∇u,∇u〉=2$〈a_{ij}(x)\mathrm{\nabla }u,\mathrm{\nabla }u〉=2$ on ∂{u>0}$\partial \{u>0\}$. We use a singular perturbation framework to approximate this free boundary problem by regularizing ones of the form: Luε=βε(uε)$L{u}_{\epsilon }={\beta }_{\epsilon }(u_{\epsilon })$, where βε${\beta }_{\epsilon }$ is a suitable approximation of Dirac delta function δ₀${\delta }_0$. A useful variational characterization to solutions of the above approximating problem is established and used to obtain important geometric properties that enable regularity of the free boundary. This theory has been developed in connection to a very recent line of research as an effort to study existence and regularity theory for free boundary problems with gradient dependence upon the penalization.     

Convexity estimates for nonlinear elliptic equations and application to free boundary problems

We prove the convexity of the set which is delimited by the free boundary corresponding to a quasi-linear elliptic equation in a 2-dimensional convex domain. The method relies on the study of the curvature of the level lines at the points which realize the maximum of the normal derivative at a given level, for analytic solutions of fully nonlinear elliptic equations. The method also provides an estimate of the gradient in terms of the minimum of the (signed) curvature of the boundary of the domain, which is not necessarily assumed to be convex.     

Solvability,asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω⁺) is embedded in an unbounded fluid domain (Ω^?). The corresponding physical process is described by the boundary‐transmission problem for second‐order partial differential equations. In particular, in the bounded domain Ω⁺, we have a 4×4 dimensional matrix strongly elliptic second‐order partial differential equation, while in the unbounded complement domain Ω^?, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.