We consider the stationary flow of a generalized Newtonian fluid which is modelled by an anisotropic dissipative potential $f$. More precisely, we are looking for a solution $u:\\Omega \\rightarrow \\mathbb{R}^n$, $\\Omega \\subset \\mathbb{R}^n$, $n=2,3$, of the following system of nonlinear partial differential equations $$ \\left\\{\\begin{aligned}-\\text{div} \\{T(\\varepsilon (u))\\}+u^k\\frac{\\partial u}{\\partial x_k}+\\nabla \\pi &=g \\quad \\text{in}\\ \\Omega ,\\\\ \\text{div}\\ u=0 \\quad \\text{in}\\ \\Omega, \\qquad u&=0 \\quad \\text{on}\\\\partial \\Omega.\\end{aligned}\\right. \\eqno{(*)} $$ Here $\\pi :\\Omega \\rightarrow \\mathbb{R}$ denotes the pressure, $g$ is a system of volume forces, and the tensor $T$ is the gradient of the potential $f$. Our main hypothesis imposed on $f$ is the existence of exponents $1

0 $. Under natural assumptions on $p$ and $q_0$ we prove the existence of a weak solution $u$ to the problem (*), moreover we prove interior $C^{1,\\alpha }$-regularity of $u$ in the two-dimensional case. If $n=3$, then interior partial regularity is established.

• Thematic program: Nonlinear Analysis (2005)

We discuss a possible issue to solve the isoperimetric problem for the buckling load of a clamped plate. This is to prove that the ball is solution of the following $$ \\min _{\\Omega \\in \\R^2, |\\Omega| =c} \\min _{u \\in H^2_0 (\\Omega), u \\not= 0} \\frac{\\int_\\Omega |\\Delta u|^2 dx}{\\int_\\Omega |\\nabla u|^2 dx}.$$ In 1995 Willms and Weinberger proved that if this problem has a solution, which is a simply connected smooth set, then this solution is the ball. In this joint work with Mark Ashbaugh, we prove the existence of a solution and discuss the minimal necessary regularity in order to solve the conjecture.

• Thematic program: Nonlinear Analysis (2005)

We will consider the following minimization problem: For given domain $D\\subset R^n$, minimize the functional $$ J(v)=\\int_{D} (\\frac{|\\nabla v|^{2}}{2}+v^{+})$$ over the space $K=\\{v\\in W^{1,2}(D): v-u_0 \\in W^{1,2}_0(D)\\}$ for a given $u_0 \\in W^{1,2}(D)$. It is important to note that $u_0$ can take negative values. We give a review of the problem and a classification of the global solutions.

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)
• WK seminar

• Thematic program: Nonlinear Analysis (2005)

I will discuss questions concerning the geometry of Laplacian growth, including some recent ideas due to Krichever, Marshakov, Mineev-Weinstein, Wiegmann, Zabrodin and others on more general Laplacian deformations.

• Thematic program: Nonlinear Analysis (2005)

We discuss and construct an action functional (logarithmic action)for simply connected Laplacian growth and obtain its variation. This variation admits various interpretations. In particular, it represents the infinitesimal version of the action of the Virasoro-Bott group over the space of analytic univalent functions.

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)

• Thematic program: Nonlinear Analysis (2005)