 Solution

The range of $v\in C^0(\overline{B})$ is compact. Continuity and positivity of $\lambda ,\Lambda$ therefore give constantsThe coefficients ${\alpha }_{ij}(x)=a_{ij}(x,v(x))$ meet the hypotheses of part (a), which gives a unique weak solution. Its global De Giorgi–Nash–Moser estimate gives $u\in C^{0,\mu }(\overline{B})$ for some $\mu =\mu (v)\in (0,1)$.