7 problems

Consider a bounded regular domain $\Omega\subset\mathbb{R}^n$ and the divergence-form equation … Here the coefficients $a^{ij}$ are bounded and measurable, and $u$ is a local weak…

Let $B$ be a ball, let $p>1$, and let $u\in W^{1,r}(B)$ with $r>p-1$. Suppose that $u$ solves … in the distributional sense. Iwaniec-Sbordone conjecture. Then…

Let $M$ be a Riemannian manifold, let $E$ be a Hermitian vector bundle over $M$, and let $V$ be a potential satisfying Assumption A: $V=V_{+}+V_{-}$, with $V_{+}(x)8e0$ and…

Let $\Omega$ be the domain and consider the coupled elliptic system with positive exponents $p_1,p_2$, where the critical hyperbola condition is … A positive weak solution is a wea…

Let $p\in(2,\infty)$, let $B_1$ be the unit ball, and let $\Delta_p u\coloneqq\operatorname{div}(|Du|^{p-2}Du)$. A function is locally of class $C^{1,\alpha}$ if it has locally Höl…

Let $p>1$, $n\geq 2$, and $r>\max\{p-1,1\}$. Let $u\in W^{1,r}_0(\mathbb{B}^n)\cap W^{1,p}_0(\mathbb{B}^n)$ and $f\in L^{\frac{r}{p-1}}\cap L^{p'}(\mathbb{B}^n,\mathbb{R}^n)$, wher…

Let $u_0$ be the limiting map in Theorem 1 and let $\psi_0$ be the function associated with $u_0$ there, satisfying the stated matrix-valued elliptic equation and boundary conditio…