Yang-Mills fields on CR manifolds

We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR manifold.Comment: 52 page

Levi umbilical surfaces in complex space

We define a complex connection on a real hypersurface of \C^{n+1} which is naturally inherited from the ambient space. Using a system of Codazzi-type equations, we classify connected real hypersurfaces in \C^{n+1}, $n\ge 2$, which are Levi umbilical and have non zero constant Levi curvature. It turns out that such surfaces are contained either in a sphere or in the boundary of a complex tube domain with spherical section.Comment: 18 page

The Deformation of an Elastic Substrate by a Three-Phase Contact Line

Young's classic analysis of the equilibrium of a three-phase contact line ignores the out-of-plane component of the liquid-vapor surface tension. While it has long been appreciated that this unresolved force must be balanced by elastic deformation of the solid substrate, a definitive analysis has remained elusive because conventional idealizations of the substrate imply a divergence of stress at the contact line. While a number of theories of have been presented to cut off the divergence, none of them have provided reasonable agreement with experimental data. We measure surface and bulk deformation of a thin elastic film near a three-phase contact line using fluorescence confocal microscopy. The out-of-plane deformation is well fit by a linear elastic theory incorporating an out-of-plane restoring force due to the surface tension of the gel. This theory predicts that the deformation profile near the contact line is scale-free and independent of the substrate elastic modulus.Comment: 4 pages, 3 figure

Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains

We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +\epsilon$. The result, in particular, implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in $L^p$ for (3/2)-\varep < p< 3+\epsilon. This gives an affirmative answer to a conjecture of M. Taylor.Comment: 28 page. Minor revision was made regarding the definition of the Stokes operator in Lipschitz domain

Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields

We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and Poincar\'e inequality for nonsmooth H\"ormander vector fields.Comment: arXiv admin note: material from arXiv:1106.2410v1, now three separate articles arXiv:1106.2410v2, arXiv:1201.5228, arXiv:1201.520

Analyticity of layer potentials and L2L^{2} solvability of boundary value problems for divergence form elliptic equations with complex L∞L^{\infty} coefficients

We consider divergence form elliptic operators of the form L=-\dv A(x)\nabla, defined in $R^{n+1} = \{(x,t)\in R^n \times R \}$, $n \geq 2$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic, complex and $t$-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on $L^2(\mathbb{R}^{n})=L^2(\partial\mathbb{R}_{+}^{n+1})$, is stable under complex, $L^{\infty}$ perturbations of the coefficient matrix. Using a variant of the $Tb$ Theorem, we also prove that the layer potentials are bounded and invertible on $L^2(\mathbb{R}^n)$ whenever $A(x)$ is real and symmetric (and thus, by our stability result, also when $A$ is complex, $\Vert A-A^0\Vert_{\infty}$ is small enough and $A^0$ is real, symmetric, $L^{\infty}$ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with $L^2$ (resp. $\dot{L}^2_1)$ data, for small complex perturbations of a real symmetric matrix. Previously, $L^2$ solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients $A_{j,n+1}=0=A_{n+1,j}$, $1\leq j\leq n$, which corresponds to the Kato square root problem