Bounds on the negative eigenvalues of Laplacians on finite metric graphs

For a self--adjoint Laplace operator on a finite, not necessarily compact, metric graph lower and upper bounds on each of the negative eigenvalues are derived. For compact finite metric graphs Poincar\'{e} type inequalities are given.Comment: 17 page

Sign-indefinite second order differential operators on finite metric graphs

The question of self-adjoint realizations of sign-indefinite second order differential operators is discussed in terms of a model problem. Operators of the type -\frac{d}{dx} \sgn (x) \frac{d}{dx} are generalized to finite, not necessarily compact, metric graphs. All self-adjoint realizations are parametrized using methods from extension theory. The spectral and scattering theory of the self-adjoint realizations are studied in detail.Comment: 43 pages, 2 figure

Maximal LpL^p-regularity and H∞H^{\infty}-calculus for block operator matrices and applications

Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part with $\mathsf{D}(\mathcal{A})=\mathsf{D}(A)\times \mathsf{D}(D)$ though with possibly large relative bound. For such operators the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.Comment: 55 pages. Accepted for publication in JF

Analyticity of solutions to the primitive equations

This article presents the maximal regularity approach to the primitive equations. It is proved that the $3D$ primitive equations on cylindrical domains admit a unique, global strong solution for initial data lying in the critical solonoidal Besov space $B^{2/p}_{pq}$ for $p,q\in (1,\infty)$ with $1/p+1/q \leq 1$. This solution regularize instantaneously and becomes even real analytic for $t>0$.Comment: 19 page

The primitive equations with stochastic wind driven boundary conditions: global strong well-posedness in critical spaces

This article studies the primitive equations for geophysical flows subject to stochastic wind driven boundary conditions modeled by a cylindrical Wiener process. A rigorous treatment of stochastic boundary conditions yields that these equations admit a unique global, strong, pathwise solution within the $L^q_t$-$L^p_x$-setting of critical spaces. Critical spaces are established for the first time within the setting of the stochastic primitive equations.Comment: 20 page