We give a comprehensive study of the 3D Navier–Stokes–Brinkman–Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the corresponding uniform attractors

We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

We examine the elliptic system given by

$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill & {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill & {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$

$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$

$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$

We study a form of optimal transportation surplus functions which arise in hedonicpricing models. We derive a formula for the Ma–Trudinger–Wang curvature of thesefunctions, yielding necessary and sufficient conditions for them to satisfy(A3w). We use this to give explicit new examples of surplus functionssatisfying (A3w), of the formb(x,y) = H(x + y)where H is a convex function on ℝn. We alsoshow that the distribution of equilibrium contracts in this hedonic pricing model isabsolutely continuous with respect to Lebesgue measure, implying that buyers are fullyseparated by the contracts they sign, a result of potential economic interest.

We apply Robin penalization to Dirichlet optimal control problemsgoverned by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.

The boundary control problem for the dynamical Lame system(isotropic elasticity model) is considered. The continuity ofthe “input → state" map in L 2-norms is established. A structure of thereachable sets for arbitrary T>0 is studied.In general case, only the first component $u(\cdot ,T)$ of thecomplete state $\{ u(\cdot ,T),u_t(\cdot ,T)\}$ may be controlled, an approximate controllability occurring inthe subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundarydata continuation. If T 0 exceeds the time neededfor shear waves to fill the entire domain, then the responseoperator (“input → output" map) $R^{2T_0}$ uniquely determinesRT for any T>0. A procedure recovering R ∞via $R^{2T_0}$ is also described.