= Solution
Write $u=\varphi+w$ with $w\in W_0^{1,2}(B)$. On this space define
$$
\mathcal B(w,z)=\int_B\alpha_{ij}D_jwD_i z.
$$
Boundedness of the coefficients makes $\mathcal B$ bounded, and ellipticity gives
$$
\mathcal B(w,w)\geq\gamma\lVert Dw\rVert_2^2,
$$
which is coercive by the <Poincare inequality>. The functional $z\mapsto-\mathcal B(\varphi,z)$ is bounded, so the <Lax-Milgram theorem> gives a unique $w$ and hence a unique weak solution $u$.
The supplied global <De Giorgi-Nash-Moser theorem> and the weak maximum principle give $u\in C^{0,\beta}(\overline B)$ for some $\beta=\beta(n,\gamma,\Gamma)$. Finally, test the weak equation with $u\psi^2$. Ellipticity, the coefficient bound, and <Young inequality> yield
$$
\gamma\int_B|Du|^2\psi^2
\leq2\Gamma\int_B|u||Du||\psi||D\psi|
\leq\frac\gamma2\int_B|Du|^2\psi^2
+\frac{2\Gamma^2}{\gamma}\int_Bu^2|D\psi|^2.
$$
Rearranging gives
$$
\boxed{
\int_B|Du|^2\psi^2
\leq4\left(\frac\Gamma\gamma\right)^2
\int_Bu^2|D\psi|^2.}
$$
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