FROM POINCARÉ TO CAUCHY–SCHWARZ INEQUALITIES: A UNIFIED APPROACH TO PDE SOLUTIONS

We study energy estimates for classical and fractional Dirichlet problems through a functional setting that records the regularity of a solution and its nonlinear image. For the spectral and restricted Dirichlet fractional Laplacians on bounded Lipschitz domains, we equip the domain $ \{u\in D(B^{1/2}):f(u)\in D(B^{1/2})\} $ with a complete graph metric whenever $ f $ is continuous and fixes zero. The associated squared functional is lower semicontinuous in $ L^2 $ and has compact sublevel sets. For the restricted operator of order $ 0<\alpha<2 $ and the power map $ f_p(t)=|t|^{p-1}t $, with $ p>1 $ and $ n>\alpha $, we establish a Hölder inclusion into $ L^{q_*} $, where $ q_*=2np/(n-\alpha) $, and compactness below this exponent. A concentration sequence proves optimality of both thresholds. Dirichlet form estimates and power dissipation identify the nonlinear transformation whose fractional regularity is controlled by an absorption equation. Fractional wave energies are treated under an explicit potential coercivity condition. Supporting results include variational estimates, parabolic regularity in the Dirichlet operator domain, and stability and second-order convergence of a classical wave discretization under a fixed strict CFL condition and smooth-solution assumptions.