# Second order elliptic PDE problem with boundary conditions whose solutions depend continuously on the initial data

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 $$U$$.

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).