We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature ${ℳ}_p^{\alpha }\left(X\right):={\int }_X{\int }_X{\int }_X{\kappa }^p(x,y,z)d_X^{\alpha }\left(x\right)d_X^{\alpha }\left(y\right)d_X^{\alpha }\left(z\right)$, where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that ${ℳ}_p^{\alpha }\left(X\right)<\infty$ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant case p = 3 for...

We incorporate model uncertainty into a quadratic portfolio optimization framework. We consider an incomplete continuous time market with a non-tradable stochastic factor. Two stochastic game problems are formulated and solved using Hamilton-Jacobi-Bellman-Isaacs equations. The proof of existence and uniqueness of a solution to the resulting semilinear PDE is also provided. The latter can be used to extend many portfolio optimization results.

We study the shape of stationary surfaces with prescribed mean curvature in the Euclidean 3-space near boundary points where Plateau boundaries meet free boundaries. In deriving asymptotic expansions at such points, we generalize known results about minimal surfaces due to G. Dziuk. The main difficulties arise from the fact that, contrary to minimal surfaces, surfaces with prescribed mean curvature do not meet the support manifold perpendicularly along their free boundary, in general.

This work is concerned with the flow of a viscous plastic fluid. We choose a model of Bingham type taking into account inhomogeneous yield limit of the fluid, which is well-adapted in the description of landslides. After setting the general threedimensional problem, the blocking property is introduced. We then focus on necessary and sufficient conditions such that blocking of the fluid occurs. The anti-plane flow in twodimensional and onedimensional cases is considered. A variational formulation...

This work is concerned with the flow of a viscous plastic fluid. We choose a model of Bingham type taking into account inhomogeneous yield limit of the fluid, which is well-adapted in the description of landslides. After setting the general threedimensional problem, the blocking property is introduced. We then focus on necessary and sufficient conditions such that blocking of the fluid occurs. The anti-plane flow in twodimensional and onedimensional cases is considered. A variational formulation...

Let $𝒴$  be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\to 𝒴$  with equibounded total variation give riseto equivalence classes of cartesian currents in ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of  $𝒴$  iscommutative. In any dimension  $n$  we prove that every element $T$  in  ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  can be approximatedweakly in the sense of currents by a sequence of graphs...

As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical...

Conditions under which the inf-convolution of f and g $f\square g\left(x\right):=in{f}_{y+z=x}(f\left(y\right)+g\left(z\right))$ has the cancellation property (i.e. f □ h ≡ g □ h implies f ≡ g) are treated in a convex analysis framework. In particular, we show that the set of strictly convex lower semicontinuous functions $f:X\to ℝ\cup +\infty$ on a reflexive Banach space such that $li{m}_{\parallel x\parallel \to \infty }f\left(x\right)/\parallel x\parallel =\infty$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.