Let $\Omega$ be a bounded subset of $\mathbf R^N$ with a $C^{2,\varepsilon}$ boundary $\partial\Omega$, $\alpha\in C^2(\overline\Omega)$ with $\alpha>0$ in $\overline{\Omega}$ and $A$ the operator defined by $Au:=\nabla\cdot (\alpha\nabla u)$ with the nonlinear general Wentzell boundary condition $$ Au+b\frac{\partial u}{\partial n}\in c\, \beta(\cdot, u)\quad\text{on}\quad\partial \Omega, $$ where $n(x)$ is the unit outer normal at $x$, $b,\, c$ are real-valued functions in $C^1(\partial\Omega)$ and $\beta (x,\cdot)$ is a maximal monotone graph. Then, under additional assumptions on $b, c, \beta$, we prove the existence of a contraction semigroup generated by the closure of $A$ on suitable $L^p$ spaces, $1\le p < \infty$ and on $C(\overline{\Omega}).$ Questions involving regularity are settled optimally, using De Giorgi-Nash-Moser iteration.

Let $(M,g)$ be a smooth compact Riemannian $n$-manifold, $n \ge 3$. Let also $p \ge 1$ be an integer, and $M_p^s({\mathbb R})$ be the vector space of symmetrical $p\times p$ real matrices. For $A: M \to M_p^s({\mathbb R})$ smooth, $A = (A_{ij})$, we consider vector-valued equations, or systems, like $$\Delta_g^p{\mathcal U} + A(x){\mathcal U} = \frac{1}{2^\star}D_{{\mathcal U}}\vert{\mathcal U}\vert^{2^\star}, $$ where ${\mathcal U}: M \to {\mathbb R}^p$ is a $p$-map, $\Delta_g^p$ is the Laplace-Beltrami operator acting on $p$-maps, and $2^\star$ is critical from the Sobolev viewpoint. We investigate various questions for this equation, like the existence of minimizing solutions, the existence of high energy solutions, blow-up theory, and compactness. We provide the complete $H_1^2$-theory for blow-up, sharp pointwise estimates, and prove compactness when the equations are not trivially coupled and of geometric type.