Partial regularity of solutions to model Venttsel problem for quasilinear elliptic systems of equations with quadratic nonlinearity relative to the gradient
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, Hölder continuity of the solution is proved in a neighborhood of a boundary point and the possible singular set of solutions is characterized as well.
scientific article; zbMATH DE number 6532821