We investigate the uniqueness of symmetric weak solutions to the stationary Navier-Stokes equation in a two-dimensional exterior domain $\Omega$. It is known that, under suitable symmetry condition on the domain and the data, the problem admits at least one symmetric weak solution tending to zero at infinity. Given two symmetric weak solutions $u$ and $v$, we show that if $u$ satisfies the energy inequality $\| \nabla u \|_{L^2 (\Omega)}^2 \le (f,u)$ and $\sup_{x \in \Omega} (|x|+1)|v(x)|$ is sufficiently small, then $u=v$. The proof relies upon a density property for the solenoidal vector field and the Hardy inequality for symmetric functions.

In this paper, we first identify some integrability and regularity issues that frequently occur in fractional calculus of variations. In particular, it is well-known that Riemann-Liouville derivatives make boundary singularities emerge. The major aim of this paper is to provide a framework ensuring the validity of the fractional Euler-Lagrange equation in the case of a Riemann-Liouville derivative of order $\alpha \in (0,1)$. For this purpose, we consider the set of functions possessing $p$-integrable Riemann-Liouville derivatives and we introduce a class of quasi-polynomially controlled growth Lagrangian. In the first part of the paper, we prove a new fractional fundamental (du Bois-Reymond) lemma and a new fractional integration by parts formula involving boundary terms. The proof of the second result is based on an integral representation of functions possessing Riemann-Liouville derivatives. In the second part of the paper, we give not only a necessary optimality condition of Euler-Lagrange type for fractional Bolza functionals, but also necessary optimality boundary conditions. Finally, we give an additional application of our results: we prove an existence result for solutions of linear fractional boundary value problems. This last result is based on a Hilbert structure and the classical Stampacchia theorem.

In the present paper, we study existence of nontrivial weak solutions for a class of Hamiltonian systems of type \begin{align} \left \{ \begin{array}{ll} -\Delta u +b(x) u = w_{1}(x)f_{1}(v), & v > 0; \\ -\Delta v +b(x) v =w_{2}(x)f_{2}(u), & u > 0 , \end{array} \right . \end{align} where $b:\mathbb{R}^2 \rightarrow \mathbb{R}$ is a continuous potential which may change sign and the nonlinearity $f_{i}:\mathbb{R} \rightarrow \mathbb{R}$ has critical or subcritical exponential growth in the sense of Trudinger-Moser's inequality, for $i=1,2$. The main results are proved by using variational methods through strongly indefinite functionals.

We investigate the following singular quasilinear elliptic system, \begin{equation} \nonumber \tag{P} \left. \begin{alignedat}{2} - \Delta_p u & = \frac{1}{ u^{a_1} v^{b_1} }\, \quad\mbox{ in }\,\Omega ;\quad u\vert_{\partial\Omega} & = 0 ,\quad u > 0\quad\mbox{ in }\,\Omega , \\ - \Delta_q v & = \frac{1}{ v^{a_2} u^{b_2} }\, \quad\mbox{ in }\,\Omega ;\quad v\vert_{\partial\Omega} & = 0 ,\quad v > 0\quad\mbox{ in }\,\Omega , \end{alignedat} \quad \right\} \end{equation} where $\Omega$ is an open bounded domain with smooth boundary, $1 < p, q < \infty$, and the numbers $a_1, a_2, b_1, b_2 > 0$ satisfy certain upper bounds. We employ monotonicity methods in order to prove the existence and uniqueness of a pair of positive solutions to (P). While following a standard fixed point approach with ordered pairs of sub- and super solutions, we need to prove a new regularity result of independent interest for solution pairs to problem (P) in $C^{0,\beta}(\overline{\Omega})$ with some $\beta\in (0,1)$.

In this paper, we investigate linear elliptic, second-order boundary value problems with mixed boundary conditions on domains with a rough boundary. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove first Hölder continuity of the solution. Secondly, Gaussian Hölder estimates for the corresponding heat kernel are derived.

We establish higher-order weighted Sobolev and Hölder regularity for solutions to variational equations defined by the elliptic Heston operator, a linear second-order degenerate-elliptic operator arising in mathematical finance [27]. Furthermore, given $C^\infty$-smooth data, we prove $C^\infty$-regularity of solutions up to the portion of the boundary where the operator is degenerate. In mathematical finance, solutions to obstacle problems for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.