The following is an excerpt from the Partial Differential Equations by Evans (2nd edition chapter 6 p.314 ):
In (7), we have the assumption that $$ a^{ij}, b^i,c\in L^\infty(U),\quad f\in L^2(U). $$
Here are my questions:
- Would anybody explain how (7) is true for any $v\in H_0^1(U)$?
- In the book, the author only assumes that $U$ is an open, bounded subset of $\mathbb{R}^n$. Do we need extra assumptions for $\partial U$?
Suppose $v\in H_0^1(U)$. Then by definition of $H_0^1(U)$, there exists $(v_m)$ with $$ v_m\to v\quad\textrm{in }H_0^1(\Omega). $$ How shall I estimate $$ \left|\int_U\sum a^{ij}u_{x_i}(v_m-v)_{x_j}+\sum b^i u_{x_i}(v_m-v)+cu(v_m-v)-f(v_m-v)\ dx\right|\ ? $$
If $a^{ij},b^j,c\in L^2(U)$, then one can use Cauchy-Schwartz inequality in the Hilbert space $H^1(U)$ to do the estimate. I don't know how to do it in this setting.
At the beginning of this section, Evans makes the assumptions $$ a^{ij},b^i,c\in L^\infty(U), f\in L^2(U). $$ The following inequalities are crucial: if $w\in H^1(U)$, then $$ \|w\|_{L^2(U)} \leq \|w\|_{H^1(U)}, \|w_{x_i}\|_{L^2(U)} \leq \|w\|_{H^1(U)}. $$ This is pretty much immediate from the definition of the Sobolev norm. You may need to tack on multiplicative constants in each inequality depending on which one of the equivalent definitions of Sobolev norm you are using.
With this, if $u\in H_0^1(U)$, $v\in H_0^1(U)$, and $v_n\in C_c^\infty(U)$ approximates $v$ in $H^1(U)$-norm, we consider the following terms (using Einstein summation convention): $$ \int_U a^{ij}u_{x_i}(v_{x_j}-v_{n_{x_j}}) + b^iu_{x_i}(v-v_n) + cu(v-v_n) = (1) + (2) + (3). $$ Each term can be estimated as follows using Hölder's inequality: $$ |(1)| \leq \|a^{ij}\|_{L^\infty(U)}\|u_{x_i}\|_{L^2(U)}\|v_{x_j} - v_{n_{x_j}}\|_{L^2(U)} \leq \|a^{ij}\|_{L^\infty(U)}\|u\|_{H^1(U)}\|v-v_n\|_{H^1(U)}, $$ $$ |(2)| \leq \|b^i\|_{L^\infty(U)}\|u_{x_i}\|_{L^2(U)}\|v-v_n\|_{L^2(U)} \leq \|b^i\|_{L^\infty(U)}\|u\|_{H^1(U)}\|v-v_n\|_{H^1(U)}, $$ $$ |(3)| \leq \|c\|_{L^\infty(U)}\|u\|_{L^2(U)}\|v-v_n\|_{L^2(U)} \leq \|c\|_{L^\infty(U)}\|u\|_{H^1(U)}\|v-v_n\|_{H^1(U)}. $$ Also, $$ \left|\int f(v-v_n)\right| \leq \|f\|_{L^2(U)}\|v-v_n\|_{L^2(U)} \leq \|f\|_{L^2(U)}\|v-v_n\|_{H^1(U)}. $$ Since $\|v-v_n\|_{H^1(U)}\to 0$ as $n\to\infty$, and every other norm in these inequalities is finite by assumption, we thus conclude that $$ \int_U a^{ij}u_{x_i}v_{x_j} + b^iu_{x_i}v + cuv = \int_U fuv $$ for any $v\in H_0^1(U)$. We can see also that Evans has pretty much chosen the minimal conditions on the coefficients $a^{ij},b^i,c$, and the inhomogeneous term $f$ for these above integrals to exist.
How much regularity is required on $U$ for this to work? None at all other than open bounded, since all we need is that every $v\in H_0^1(U)$ has an approximating sequence in $C_c^\infty(U)$, and since $H_0^1(U)$ is the closure of $C_c^\infty(U)$ in $H^1$-norm this comes for free. Additional assumptions on $\partial U$ give you extension theorems, trace theorems, and additional Sobolev estimates (see Chapter 5), but none of these are required for the above estimates to hold. (Edit: We do need integration by parts to work to get to the integral formulation in the first place. So some regularity on $\partial U$ would be required, like piecewise Lipschitz.)