# Founding bounds on a certain expression

Suppose we have a bounded domain $$\Omega$$ with a boundary $$\Gamma$$. The space $$L^2(\Omega)$$ is equipped with the usual norm and inner product $$|| \cdot||$$ and $$(\cdot , \cdot)$$. I was working on a problem, and we want to find bounds on $$L_E$$ where: $$L_E = (f,\dot{u}) + (g,\dot{u})_{\Gamma} + (\dot{w}, \varphi) + (\dot{p}, \varphi)_{\Gamma} + (Q, \theta) - (q, \theta)_{\Gamma}$$ By using Holder's and Young's inequalities and trace theorem I found: $$L_E \leq C + \frac{C'}{2} \left[||\dot{u}||^2 + ||\nabla \dot{u}||^2 + ||\varphi||^2 + ||\nabla \varphi||^2 + ||\theta||^2 + ||\nabla \theta||^2 \right]$$ Where $$C'$$ is a positive constant (that comes from the trace inequality constants) and: $$C = \frac{1}{2} \left[||f||^2 + ||g||^2_{\Gamma} + ||\dot{w}||^2 + ||\dot{p}||^2_{\Gamma} + ||Q||^2 + ||q||^2_{\Gamma} \right]$$ But in this article I am reading they found: $$L_E \leq C + \frac{1}{2} \left[||\dot{u}||^2 + ||\varphi||^2 + ||\theta||^2 \right]$$ Where did I go wrong? Or what am I missing?

• Some context would be helpful. It's not even clear what things like $(\cdot,\cdot)_\Gamma$ or $\dot{u}$ mean. Apr 3, 2022 at 8:58
• $(\cdot, \cdot)_{\Gamma}$ is the inner product at the boundary $\Gamma$ i.e. $(w,v)_{\Gamma} = \int \limits_{\Gamma} w v dx$ and $\dot{u}$ is the derivative of $u$ with respect to time. Apr 3, 2022 at 12:50