Partial regularity of solutions to model Venttsel problem for quasilinear elliptic systems of equations with quadratic nonlinearity relative to the gradient
type: Item
instance of: scholarly article
title: Partial regularity of solutions to model Venttsel problem for quasilinear elliptic systems of equations with quadratic nonlinearity relative to the gradient
zbMATH Open document ID: 1336.35150
published in: Journal of Mathematical Sciences (New York)
publication date: 2016-01-19T00:00:00Z
review text: The very interesting paper under review deals with regularity issues regarding the weak solution to a Venttsel problem for quasilinear elliptic systems with quadratic gradient nonlinearity. More precisely, the author considers the problem \[ \begin{cases} -\operatorname{div} \big(a(x,u)\nabla u\big)=f(x,u,\nabla u) & x\in B_1^+(0),\\ -\operatorname{div}' \big(b(x',u)\nabla' u\big)+(a\nabla u,\mathbf{n})=g(x',u,\nabla' u) & x'\in \Gamma_1(0), \end{cases} \] where \(x=(x',x_n)\), \(x'=(x_1,\dots,x_{n-1})\), \(B_1^+(0)=B_1(0)\cap \{x_n>0\}\), \(B_1(0)=\left\{x\in\mathbb R^n:| x|<1\right\}\), \(n\geq 2\), \(u: B_1^+(0)\to \mathbb R^N\), \(N>1\), \(u=(u^1,\dots,u^N)\), \(\nabla 'u=\left\{u^k_{x_i}\right\}_{i\leq n-1}^{k\leq N}\), \[ \operatorname{div}'(b\nabla 'u)=\sum_{i=1}^{n-1} (bu_{x_i})_{x_i}, \] and \(\mathbf{n}=(0,\dots,0,-1)\) is the unit outward normal to the surface \(\Gamma_1(0)=B_1(0)\cap \{x_n=0\}\). The matrices \(a(x,u)=\left\{a_{kl}^{\alpha\beta}(x,u)\right\}_{k,l\leq N}^{\alpha,\beta\leq n}\) and \(b(x',u)=\left\{b_{kl}^{\tau s}(x',u)\right\}_{k,l\leq N}^{\tau,s\leq n-1}\) are supposed to be elliptic, continuous in \(u\) and VMO in \(x\), (\(x'\) respectively), while the free terms \(f\) and \(g\) support quadratic gradient growths, \[ \left| f(x,u,p)\right|\leq K_f | p|^2 +f_0(x)\quad x\in B_1^+(0), \;u\in \mathbb R^N, \;p\in \mathbb R^{nN}, \] \[ \left| g(x',u,q)\right|\leq K_g | q|^2 +g_0(x')\quad x'\in \Gamma_1(0), \;u\in \mathbb R^N, \;p\in \mathbb R^{(n-1)N} \] with constants \(K_f\) and \(K_g\), and \(f_0\in L^2(B_1^+(0))\), \(g_0\in L^2(\Gamma_1(0))\). Employing the method of the \(A\)-harmonic approximation [the author, ibid. 198, No. 6, 655--676 (2014; Zbl 1307.35114); translation from Probl. Mat. Anal. 75, 3--20 (2014)], the author proves partial regularity of the weak solution in a neighborhood of \(\Gamma_1(0)\) by describing preliminary conditions which guarantee local Hölder continuity of the solution along this boundary part. After that, …
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J48
Mathematics Subject Classification ID: 35J66
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35J62
zbMATH DE Number: 6532821
zbMATH Keywords: quasilinear elliptic system
zbMATH Keywords: Venttsel boundary condition
zbMATH Keywords: partial regularity
zbMATH Keywords: Hölder continuity
zbMATH Keywords: singular set
author: Arina A. Arkhipova Arkhipova
MaRDI profile type: Publication
full work available at URL: s10958-015-2579-y
OpenAlex ID: W1959280446
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DOI: 10.1007/S10958-015-2579-Y
description: scientific article; zbMATH DE number 6532821