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My question refers to differential equations in Sobolev spaces. It is as follows:

Let $\Omega \subset \mathbb{R}^n$ be a bounded open set. Let $a: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow \mathbb{R}$ be a bilinear symmetric form, such that:

  1. (continuity in $H^1$) $\forall u,v \in H^1(\Omega) \; \; \; a(u,v) \leq c_1 \|u\|_{H^1(\Omega)} \|v\|_{H^1(\Omega)}$,
  2. ($L_2$ coercivity in $H^1_0$) $\forall \eta \in H_0^1(\Omega) \; \; \; a(\eta,\eta) \geq c_2 \|\eta\|_{L_2(\Omega)}^2$.

Let $u,w$ be solutions of equations $$ a(u,v) = \int_{\Omega} g(x)v(x) dx \quad \forall v \in H_0^1(\Omega); u|_{\partial \Omega} = \bar{u}|_{\partial \Omega},$$ $$ a(w,v) = \int_{\Omega} g(x)v(x) dx \quad \forall v \in H_0^1(\Omega); w|_{\partial \Omega} = \bar{w}|_{\partial \Omega},$$ where $\bar{u}, \bar{w} \in \mathcal{C}^2(\overline{\Omega})$ and $g \in L_2(\Omega)$ are given.

Let us define $\varphi := u - \bar{u}$, $\psi := w - \bar{w}$.

My question is:

Under the following assumptions, is it true that $$a(\varphi - \psi, \varphi - \psi) \leq c_3 \|\bar{u} - \bar{w}\|_{H^1(\Omega)}^2.$$


Rationale:

If we pick some $a$, for example $a(u,v) := \int_\Omega \nabla u \cdot \nabla v \; dx$, then $u$ and $w$ may be treated as weak solutions of a given differential equation with a Dirichlet boundary values determined by the functions $\bar{u}$, $\bar{w}$. The aim of this question is to estimate the difference between these solutions in some manner.

This result seems easy if the coercivity in assumption 2 will be in $\|\cdot\|_{H^1(\Omega)}$ or even in $|\cdot|_{H_0^1(\Omega)}$ instead of $\|\cdot\|_{L_2(\Omega)}$. Then we could take $v = \varphi - \psi$ in both equations, subtract them and use assumptions 1,2 to obtain the above estimate.

Is the coercivity in $L_2$ norm enough?

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