= Weak Robin problem for a uniformly elliptic operator
For $Lu=-\partial_j(a^{ij}\partial_i u)+b^i\partial_i u+cu$ with smooth coefficients on a bounded smooth domain, the <weak formulation> of $(L+\lambda)u=f$ and conormal <Robin boundary condition> $a^{ij}\partial_i u\,\nu_j+\beta u=0$ is
$$
\int_U\bigl(a^{ij}\partial_i u\partial_j v+b^i\partial_i u\,v+(c+\lambda)uv\bigr)
+\int_{\partial U}\beta TuTv=\int_Ufv\qquad(v\in H^1(U)).
$$
The <bilinear form> is bounded by the <Sobolev trace theorem>. If $a^{ij}$ satisfy the positive <uniformly elliptic operator> inequality, the <multiplicative trace inequality on a bounded smooth domain> and the <Young inequality> show that the form is <coercive> when $\lambda$ is sufficiently large, even if $\beta$ and $c$ have negative parts. The <Lax-Milgram theorem> then gives a unique <weak solution> for every $f\in L^2(U)$.
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