Ytukyg

$textbf{Problem}$ Let $Omega subset mathbb{R}^n$ be open, bounded and connected with $partial Omegain C^1$. For each $i,j=1,cdots,n$, assume that $a_{ij},b_i,c in L^{infty}(Omega)$ (real valued function) , and assume that there exists a constant $mu in (0,1)$ satisfying
begin{align*}
mu vert xi vert^2leq sum_{i,j=1}^n a_{ij}xi_ixi_j leq frac{1}{mu} vert xi vert ^2 ; textrm{a.e. in } Omega, ; textrm{ for all } xiin mathbb{R^n}
end{align*}
Define
begin{align*}
Lu:=-sum_{i,j=1}^npartial_{j}(a_{ij}partial_iu)+sum_{i=1}^nb_i partial_iu+cu
end{align*}
For given functions $fin L^2(Omega)$ and $gin H^{1}(Omega)$, suppose that $u in H^1(Omega)$ is a weak solution to the following boundary value problem
begin{align*}
begin{cases}
Lu=f & textrm{ in } Omega \
u=g & textrm{ on } partial Omega
end{cases}
end{align*}
If the homogeneous boundary value problem
begin{align*}
begin{cases}
Lu=0 & textrm{ in } Omega \
u=0 & textrm{ on } partial Omega
end{cases}
end{align*}
has only trivial weak solution, then prove that there exists a constant $C>0$ (independent of $u,f$ and $g$) so that
begin{align*}
Vert u Vert_{H^1(Omega)} leq C(Vert f Vert_{L^2(Omega)}+Vert g Vert_{H^1(Omega)})
end{align*}

$textbf{Attempt}$

Take $w:=u-g $. Then, $w$ satisfies the following boundary value problem
begin{align*}
begin{cases}
Lw=f-Lg & textrm{ in } Omega \
w=0 & textrm{ on } partial Omega
end{cases}
end{align*}
We get $w in H^1_0(Omega)$. Also,
begin{align*}
Vert u Vert _{H^1(Omega)} &= Vert w+g Vert_{H^1(Omega)}\
&leq Vert w Vert_{H^1(Omega)} + Vert g Vert_{H^1(Omega)}\
end{align*}

($textbf{Update}$) We remain that $Vert w Vert_{H^1(Omega)}$ is bounded by $Vert f Vert _{L^2(Omega)}$ and $ Vert g Vert_{H^1(Omega)}$.

Define a bilinear map $B[cdot,cdot]$ from $H^1(Omega)$ to $H^1(Omega)$ by
begin{align*}
B[w,v]:= int_{Omega} sum_{i,j=1}^n a_{ij} partial_i w partial_j v +(sum_{i=1}^n b_ipartial_i w + cw) v ; dx
end{align*}
Then, we easily check that
begin{align*}
B[w,w]=int_{Omega} fw ; dx -int_{Omega} sum_{i,j=1}^n a_{ij} partial_i g partial_j w +(sum_{i=1}^n b_ipartial_i g +cg)w ; dx
end{align*}

$textbf{Note}$ We have the following properties:

begin{align*}
&(1) ; beta Vert w Vert_{H^1(Omega)}^2 leq B[w,w]+gamma Vert w Vert _{L^2(Omega)}^2 ; textrm{for some constants }beta>0, gammageq 0 \
&(2) ; Vert w Vert_{L^2(Omega)}leq C_p Vert Dw Vert_{L^2(Omega)} ; textrm{(Poincare's inequality)}\
&(3) ; Vert w Vert_{L^2(Omega)} leq Vert w Vert_{H^1(Omega)},; Vert Dw Vert _{L^2(Omega)} leq Vert w Vert_{H^1(Omega)} \
&(4) ; ableq epsilon a^2 +frac{b^2}{4epsilon} ; (a,b>0, epsilon>0) ; textrm{(Cauchy's inequality with }epsilon)
end{align*}

By using the properties and Holder's inequality, I induced
begin{align*}
beta Vert w Vert_{H^1(Omega)}^2 leq Vert f Vert_{L^2(Omega)} Vert w Vert _{L^2(Omega)} +C_1Vert g Vert_{H^1(Omega)}Vert Dw Vert_{L^2(Omega)} +C_2 Vert g Vert _{H^1(Omega)} Vert wVert_{L^2(Omega)} +gamma Vert w Vert _{L^2(Omega)}^2
end{align*}

However, I stuck $Vert w Vert_{H^1(Omega)}$ is bounded by $Vert f Vert_{L^2(Omega)}$ and $Vert g Vert_{H^1(Omega)}$ because of the last term $Vert w Vert_{L^2(Omega)}^2$

Any help is appreciated...

Thank you!

I'm not sure if you can apply said theorem directly, as certainly $Lg$ needs not lie in $L^2$ in general. However, you can replicate the argument to generalise the result to this situation. In particular the proof using a blow-up argument goes through with minimal modifications.

First observe that for any such $u, f, g,$ then by standard elliptic estimates (testing against $v = u-g$) there exists $C > 0$ independent of $u,f,g$ such that,
$$ lVert u rVert_{H^1(Omega)} leq Cleft( lVert u rVert_{L^2(Omega)} + lVert f rVert_{L^2(Omega)} + lVert g rVert_{H^1(Omega)}right). $$
So it suffices to find $C>0$ such that,
$$ lVert u rVert_{L^2(Omega)} leq Cleft( lVert f rVert_{L^2(Omega)} + lVert g rVert_{H^1(Omega)}right). $$
So how do we do this? Suppose otherwise, so we assume any solution to the homogenous problem,
begin{align*}
begin{cases}
Lu=0 & textrm{ in } Omega \
u=0 & textrm{ on } partial Omega
end{cases}
end{align*}
is necessarily zero. Also suppose the estimate does not hold, so we can find $u_k in H^1(Omega),$ $f_k in L^2(Omega)$ and $g_k in H^1(Omega)$ such that for each $k,$
begin{align*}
begin{cases}
Lu_k=f_k & textrm{ in } Omega \
u_k=g_k & textrm{ on } partial Omega
end{cases}
end{align*}
and we have the inequality,
$$ lVert u_k rVert_{L^2(Omega)} > k left( lVert f_k rVert_{L^2(Omega)} + lVert g_k rVert_{H^1(Omega)} right). $$
We will also assume each $lVert u_k rVert_{L^2(Omega)} = 1,$ which is always possible by rescaling. We will show this gives rise to a contradiction.

Note that we get $lVert f_k rVert_{L^2(Omega)} + lVert g_k rVert_{H^1(Omega)} leq frac1k rightarrow 0,$ so we get $f_k rightarrow 0$ in $L^2$ and $g_k rightarrow 0$ in $H^1(Omega).$ So by our earlier estimate,
$$ lVert u rVert_{H^1(Omega)} leq Cleft( lVert u rVert_{L^2(Omega)} + lVert f rVert_{L^2(Omega)} + lVert g rVert_{H^1(Omega)}right) leq Cleft(1+frac1kright) leq 2C. $$
Thus the sequence $(u_k)$ is uniformly bounded in $H^1(Omega).$ By the Rellich-Kondrachov theorem, there exists a subsequence $(u_{k_j})$ such that $u_{k_j} rightharpoonup u$ weakly in $H^1(Omega)$ and $u_{k_j} rightarrow u$ strongly in $L^2(Omega).$ We claim that $u$ solves the homogeneous problem; indeed if $v in H^1_0(Omega)$ we have,
begin{align*}
int_{Omega} Anabla u nabla v + bnabla u v + cuv ,mathrm{d}x &= lim_{jrightarrow infty} int_{Omega} Anabla u_{k_j} nabla v + bnabla u_{k_j} v + cu_{k_j}v ,mathrm{d}x \
&= lim_{j rightarrow infty} int_{Omega} f_{k_j}v ,mathrm{d}x \
&= 0.
end{align*}
Also the trace operator preserves weak limits (being linear and bounded), so $operatorname{tr} u = 0.$ Thus by the uniqueness assumption, we get $u=0$ in $Omega.$ However we also have $lVert u rVert_{L^2(Omega)} = lim_{k rightarrow infty} lVert u_{k_j} rVert_{L^2(Omega)} = 1,$ which gives our desired contradiction.

There is another approach, which might be closer to what you are looking for. For this view $L$ as a linear operator,
$$ L : H^1(Omega) rightarrow H^{-1}(Omega). $$
Then $u in H^1(Omega)$ solves the non-homogenous problem if and only if $u-g in H^1_0(Omega)$ and $L(u-g) = f - Lg.$ Hence it suffices to show that if $w in H^1_0(Omega),$ we have,
$$ lVert w rVert_{H^1(Omega)} leq C lVert Lu rVert_{H^{-1}(Omega)}. $$
Indeed if this holds, then for any solution $u$ to the non-homogenous problem, we have,
begin{align*}
lVert u rVert_{H^1(Omega)} &leq lVert u - grVert_{H^1(Omega)} + lVert g rVert_{H^1(Omega)} \
&leq C lVert f - Lg rVert_{H^{-1}(Omega)} + lVert g rVert_{H^1(Omega)} \
&leq Cleft( lVert f rVert_{L^2(Omega)} + lVert g rVert_{H^{-1}(Omega)}right).
end{align*}
In the last line we used the continuous embedding $L^2(Omega) hookrightarrow H^{-1}(Omega),$ and the fact that $L$ is bounded as an operator $H^{1}(Omega) rightarrow H^{-1}(Omega).$

So it suffices to prove an a-priori estimate for $L$ as an operator $H^1_0(Omega) rightarrow H^{-1}(Omega).$
To do this we use some functional analysis. Note this essentially amounts to doing the same thing I did above, depending on how you prove the theorems I will quote.

By Lax-Milgram, for $tau > 0$ sufficiently large we have the operator,
$$ L_{tau} = L + tau I : H^1_0(Omega) rightarrow H^{-1}(Omega) $$
is invertible (here $I$ is the inclusion $H^1_0(Omega) hookrightarrow H^{-1}(Omega)$). Now observe that for $f in H^{-1}(Omega)$ and $w in H^1_0(Omega)$ we have,
$$ Lu =f ,$$
if and only if,
$$ u - tau L^{-1}_{tau}Iu = L_{tau}^{-1}f. $$
Now as the inclusion $I : H^1_0(Omega) hookrightarrow H^{-1}(Omega)$ is compact by Rellich-Kondrachov, the operator $L^{-1}_{tau}I$ is compact. So $T = operatorname{id}_{H^{-1}(Omega)} - tau L^{-1}_{tau}I$ is a Fredholm operator. Moreover $L = L_{tau}T.$

Now by the Fredholm alternative (abstract functional analysis), we have $T$ is invertible if and only if it is injective, that is $Tu = 0$ if and only if $u = 0$ for $u in H^1_0(Omega).$ Noting $L_{tau}$ is an isomorphism, we have,
$$ L_{tau}Tu = Lu = L_{tau}0 = 0, $$
if and only if $u = 0.$ But we know this is true by assumption.

Thus for any $u in H^1_0(Omega)$ we get,
$$ lVert u rVert_{H^1_0(Omega)} leq lVert T^{-1} rVert lVert L_{tau}^{-1} rVert lVert L_{tau}Tu rVert_{H^{-1}(Omega)} = C lVert Lu rVert_{H^{-1}(Omega)}. $$

Remark: Note that in both cases, we have absolutely no idea what the constant $C$ is. In the former case a contradiction argument shows it exists, while in the latter case the constant is essentially conjured up using functional analysis magic (depending on how you prove the Fredholm alternative, there is either a contradiction argument or a Baire Category argument involved, or both).

In applications however, it is often useful to know more precise dependencies on the constants. For example subject to further assumptions (e.g. if we can apply the maximum principle), we may have $C$ only depends on things like $mu$ on the $L^{infty}$ norms of the coefficients. So often it's useful in practice to try and prove such an a-priori estimate directly, using additional information you have about your particular problem.

For a reference, see chapter 6 of the book 'Partial Differential Equations' by Evans. Although he often assumes higher regularity of his coefficients, it is not hard to modify the argument to this lower regularity setting.