Problems of stability and well-posedness in the calculus of variations and related PDEs - Kalayanamit, P (2022)
We present a collection of three closely related topics regarding the stability and well-posedness of minimization problems in the calculus of variations, namely the generic Tykhonov well-posedness with respect to linear perturbations, the generalized Hadamard well-posedness for a class of Dirichlet quasiconvex variational problems, and the $d$-rectifiability of the set of minimizers for a singularly perturbed quasiconvex variational integral. Regarding the first topic on generic Tykhonov well-posedness, we prove that by introducing a linear perturbation term $\xi$ to a variational integral of the form $J(u) = \int_{\Omega} F(x, u, \nabla u) \,dx$ satisfying a mild growth condition on the negative part of the integrand $F$, and a coercivity condition for $J$, the perturbed functional $J - \xi$ admits a unique strong minimizer in the Dirichlet class $W^{1,p}_{g}(\Omega ; \mathbb{R}^N)$ for all $\xi$ in a dense $G_\delta$-subset of $W^{1,p}_{0}(\Omega ; \mathbb{R}^N)^*$ for any $p \ge 1$. The proof for the case $p = 1$ is particularly distinctive and is based on a duality between the lower semicontinuity of $J$ on sets with the Radon-Nikodym Property and the generic differentiability of its Fenchel conjugate $J^*$. As for the second topic on generalized Hadamard well-posedness, we show that under the $p$-growth condition and the uniform strict quasiconvex condition for the integrand $F$ of $J$, the solution map $\mathcal{S} \colon g \mapsto \text{argmin}\{ J(u) : u \in W^{1,p}_{g}(\Omega ; \mathbb{R}^N) \}$ enjoys some nice semicontinuity property as a multivalued function $\mathcal{S} \colon W^{1,p}(\Omega ; \mathbb{R}^N) \rightrightarrows W^{1,p}(\Omega ; \mathbb{R}^N)$. Lastly, we study the set $\mathcal{M}$ of all the minimizers for a singularly perturbed strongly quasiconvex variational integral $J_{\varepsilon}(u) = \int_{\Omega} F(\nabla u) \,dx + \varepsilon \int_{\Omega} |\nabla^2 u|^2 \,dx$ from the Dirichlet class $W^{2,2}_{g}(\Omega ;\mathbb{R}^N)$ and show that when $F$ satisfies the $p$-growth condition for a $p\in \left[ 2,\frac{n+2}{n-2} \right)$ when $n>2$ (no condition on $p \in [1,\infty)$ for $n=2$), $\mathcal{M}$ is a $d$-rectifiable set (in Federer's sense), or, more precisely, $\mathcal{M}$ is bi-Lipschitz equivalent to a compact subset of $\mathbb{R}^d$. This result is based on an extension of a technique from a recent work by Campos Cordero and Kristensen on questions on uniqueness.