Consider the following problem

$$begin{cases}

-Delta u+cu=f,&xinOmega\

u=g,&xinpartialOmega

end{cases}$$

where $Omegasubseteqmathbb R^n$ is open with regular boundary, $cgeq0$ is a constant, $fin L^2(Omega)$ and $g$ is the trace of a function $Gin H^1(Omega)$. If we consider $u$ a weak solution to this problem, and define $U=u-Gin H_0^1(Omega)$, it is easy to see that $U$ is a weak solution to the following problem

$$begin{cases}

-Delta U+cU=f+Delta G-cG,&xinOmega\

U=0,&xinpartialOmega

end{cases}$$

It is also easy to see that we can apply Lax-Milgram theorem with the bilinear form

$$B(u,v)=int_Omegaleft(sum_{i=1}^nu_{x_i}v_{x_i}+cuvright)$$

and the bounded linear functional

$$L_f(v)=int_Omega(f-cG)v-int_Omegasum_{i=1}^n G_{x_i}v_{x_i}$$

to conclude there exists a unique weak solution $U$ to the auxiliary problem defined above. If we define $u=U+Gin H^1(Omega)$, it is clear then that this function will be a solution to the original problem.

Now to the question: I would like to prove that this solution $u$ depends continuously on the initial data, that is, that there exists a constant $C>0$ such that

$$lVert urVert_{H^1(Omega)}leq C(lVert frVert_{L^2(Omega)}+lVert GrVert_{H^1(Omega)})$$

I feel that the work I have done to prove that $L_f$ is bounded should be relevant for our purposes, because

$$lVert urVert_{H^1(Omega)}leqlVert UrVert_{H^1(Omega)}+lVert GrVert_{H^1(Omega)}$$

and

$$lVert UrVert_{H^1(Omega)}leq C B(U,U)^{1/2}= C|L_f(U)|^{1/2}$$

The problem is that I don’t know how to manipulate $L_f(U)$ to obtain the result. I have managed to prove a completely useless inequality, for it involves the norm of $U$.

I would appreciate any kind of suggestion. Thanks in advance for your answers.

**P.S.** The problem is that *a priori* $Delta G$ doesn’t have to be in $L^2(Omega)$, which makes it hard to use the $H^2$ regularity of $U$ (which would solve the problem instantly).

**P.S.S.** Also posted this question in SE.