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The boundary data are encoded by the Affine Sobolev spaceA function $u\in {\mathcal{A}}_{\phi }$ is a weak solution of the homogeneous p-Laplacian equation when

For existence, take a minimizing sequence for the p-energy on ${\mathcal{A}}_{\phi }$. The Poincare inequality bounds $u-\phi$ in $W^{1,p}$ by its gradient, so the sequence is bounded in the reflexive Banach space $W^{1,p}$. A weakly convergent subsequence remains in the weakly closed affine space, and convexity of $|\eta {|}^p$ gives weak lower semicontinuity. The direct method in the calculus of variations therefore produces a minimizer, whose first variation is precisely the displayed weak equation.

Solved by gpt-5.6-sol high.

A minimizing sequence in ${\mathrm{Lip}}_g(\Omega ,k)$ is uniformly bounded and equi-Lipschitz. The Arzela-Ascoli theorem gives a uniformly convergent subsequence with limit $u$ having the same boundary data and Lipschitz constant at most $k$. Its gradients have a weak-star convergent subsequence in $L^{\infty }$, with limit $Du$. Convexity of $F$ makes the integral functional weak-star lower semicontinuous, soThus $u$ attains the infimum by the direct method in the calculus of variations.

Solved by gpt-5.6-sol high.