# Concerning solvability of Dirichelet problem for non-uniformly elliptic equation

I am currently reading Gilbarg Trudinger's book on elliptic pde, section 6.6 on Non-uniformly elliptic equation, in theorem 6.24, it states for a *strong text*Strictly elliptic linear operator L in a bounded domain with locally holder continuous coifficients(with exponent \alpha), c <=0 , and b^{i},c,f are bounded. Suppose the domain \Omega satisfy exterior sphere condition and in addition, strictly exterior plane condition at those boundary points where any of the coifficients a^{i,j} are unbounded, then the Dirichelet problem Lu=f, u =\phi on boundary of \Omega has a unique C^{0}(\overline{\Omega}) and C^{2,\alpha} condition for arbitrary contious boundary value \phi.

It then says in Corollary 6.24, if one assumes the domain is strictly convex. The elliptic equation Lu=a^{i,j}D^{i,j}u=0 (a^{i,j} are locally holder continuous with exponent \alpha) u=\phi on boudnary of \Omega has a unique C^{0}(\overline{\Omega}) and C^{2,\alpha}(\Omega} solution for contious boundary value \phi.

Moreover, it seem to be the case that the classical dirichelet problem is also solvable if elliptic operator L=a^{i,j}D^{i,j}u+b^{i}D^{i}u where a^{i,j} are locally holder continuous with exponent \alpha,\Omega strictly convex And for each boundary point x_{0}, b.v>0 in some B(x_{0},R) intersect \Omega. Where v is the exterior normal to the supporting hyperplane pointing out from \Omega at x_{0}.

I am wondering whether one need strictly elliptic condition in Corollary 6.24 and its similar assertion or the strictly convex condition plays a role here that substitute the strictly ellipticity condition?

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