In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control theory. We show that for a class of partial differential equations, the shape-dependent state variable can be differentiated with respect to the topology, thus leading to a linearised system resembling those occurring in standard optimal control problems. However, a lot of care has to be taken when handling the regularity of the solutions of this linearised system. In fact, we should expect different notions of (very) weak solutions, depending on whether the main part of the operator or its lower order terms are being perturbed. We also study the relationship with the topological state derivative, usually obtained through classical topological expansions involving boundary layer correctors. A feature of the topological state derivative is that it can either be derived via Stampacchia-type regularity estimates or alternately with classical asymptotic expansions. It should be noted that our approach is flexible enough to cover more than the usual case of point perturbations of the domain. In particular, and in the line of (Delfour in SIAM J Control Optim 60(1):22-47, 2022; J Convex Anal 25(3):957-982, 2018), we deal with more general dilatations of shapes, thereby yielding topological derivatives with respect to curves, surfaces or hypersurfaces. To draw the connection to usual topological derivatives, which are typically expressed with an adjoint equation, we show how usual first-order topological derivatives of shape functionals can be easily computed using the topological state derivative.

The aim of this paper is to give a thorough insight into the relationship between the Rumin complex on Carnot groups and the spectral sequence obtained from the filtration on forms by homogeneous weights that computes the de Rham cohomology of the underlying group.

This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.

We show that Worm domains are not Gromov hyperbolic with respect to the Kobayashi distance.

We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in , which include Lipschitz submanifolds.

Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two-at a first sight-very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.

We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy in (Anz Österreich Akad Wiss Math Nat Kl 132:3-10, 1995). In particular, we see that the inverse of the minimal spherical dispersion is, for fixed , linear in the dimension of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse , our bounds are optimal with respect to the dependence on .

In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is -rectifiable, for , if it has positive -lower density and finite -upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare -rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of -rectifiable measures. Namely, we prove that the support of a -rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of -rectifiable measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a -rectifiable measure has almost everywhere positive and finite -density whenever the tangents admit at least one complementary subgroup.

Suppose that , , is a uniform domain with -Ahlfors regular boundary and is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in . We show that the corresponding elliptic measure is quantitatively absolutely continuous with respect to surface measure of in the sense that if and only if any bounded solution to in is -approximable for any . By -approximability of we mean that there exists a function such that and the measure with is a Carleson measure with control over the Carleson norm. As a consequence of this approximability result, we show that boundary functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy -type Carleson measure estimates with control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order . We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if . Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if , whereas if then finite-time blowup may occur. The geodesic completeness for is obtained by proving metric completeness of the space of -immersed curves with the distance induced by the Riemannian metric.

We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in can be extended to an infinitely narrow flat ribbon having bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.

In this note we prove the existence of two proper biharmonic maps between the Euclidean ball of dimension bigger than four and Euclidean spheres of appropriate dimensions. We will also show that, in low dimensions, both maps are unstable critical points of the bienergy.

This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that -positive/ -nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all .

We prove a Bochner-Kodaira-Nakano formula and establish Szegő kernel expansions on complete strictly pseudoconvex CR manifolds with transversal CR -action under certain natural geometric conditions. As a consequence we show that such manifolds are locally CR embeddable.

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being , their Cartan development in the tangent space is planar in the ordinary sense; being , their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion is totally umbilic if and only if the extrinsic shape of every geodesic of is planar. This extends a well-known result about surfaces in .

We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray-Singer analytic torsion, up to a constant.

We consider the problem of finding the best function such that for any pair of convex bodies the following Brunn-Minkowski type inequality holds where is the -convolution body of and . We prove a sharp inclusion of the family of Ball's bodies of an -concave function in its super-level sets in order to provide the best possible function in the range , characterizing the equality cases.

We obtain sharp rotation bounds for the subclass of homeomorphisms of finite distortion which have distortion function in , , and for which a Hölder continuous inverse is available. The interest in this class is partially motivated by examples arising from fluid mechanics. Our rotation bounds hereby presented improve the existing ones, for which the Hölder continuity is not assumed. We also present examples proving sharpness.

In this paper, we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank- Carnot algebra is isomorphic to the exterior algebra of . Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.

Let be an elliptically fibered 3 surface, admitting a sequence of Ricci-flat metrics collapsing the fibers. Let be a holomorphic () bundle over , stable with respect to . Given the corresponding sequence of Hermitian-Yang-Mills connections on , we prove that, if is a generic fiber, the restricted sequence converges to a flat connection . Furthermore, if the restriction is of the form for distinct points , then these points uniquely determine .

CR functions on an embedded quadric always extend holomorphically to where is the closure of the convex hull of the image of the Levi form. When is a closed polygonal cone, we show that the Bergman kernel on the interior of is a derivative of the Szegö kernel. Moreover, we develop the Hardy space theory which turns out to be particularly robust. We provide examples that show that it is unclear how to formulate a corresponding relationship between the Bergman and Szegö kernels on a wider class of quadrics.