Annali di Matematica Pura ed Applicata最新文献

The key point to prove the optimal (C^{1,frac{1}{2}}) regularity of the thin obstacle problem is that the frequency at a point of the free boundary (x_0in Gamma (u)), say (N^{x_0}(0^+,u)), satisfies the lower bound (N^{x_0}(0^+,u)ge frac{3}{2}). In this paper, we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies (W_frac{3}{2}). It allows to say that there are not (lambda -)homogeneous global solutions with (lambda in (1,frac{3}{2})), and by this frequency gap, we obtain the desired lower bound, thus a new self-contained proof of the optimal regularity.

In our previous paper, it is proved that for any positive flow-spine P of a closed, oriented 3-manifold M, there exists a unique contact structure supported by P up to isotopy. In particular, this defines a map from the set of isotopy classes of positive flow-spines of M to the set of isotopy classes of contact structures on M. In this paper, we show that this map is surjective. As a corollary, we show that any flow-spine can be deformed to a positive flow-spine by applying first and second regular moves successively.

We consider the following higher-order prescribed curvature problem on ( {mathbb {S}}^N: )

where (widetilde{K}(y)>0) is a radial function, (m^{*}=frac{2N}{N-2m}), and (D^m) is the 2m-order differential operator given by

where (g=g_{{mathbb {S}}^N}) is the Riemannian metric. We prove the existence of infinitely many double-tower type solutions, which are invariant under some non-trivial sub-groups of O(3),  and their energy can be made arbitrarily large.

In this paper we study the following class of linearly coupled systems in the plane:

where (f_{1}, f_{2}) are continuous functions with critical exponential growth in the sense of Trudinger-Moser inequality and (0<lambda <1) is a parameter. First, for any (lambda in (0,1)), by using minimization arguments and minimax estimates we prove the existence of a positive ground state solution. Moreover, we study the asymptotic behavior of these solutions when (lambda rightarrow 0^{+}). This class of systems can model phenomena in nonlinear optics and in plasma physics.

If a Lie group admits a left invariant Randers metric of scalar flag curvature, then it is called of scalar Randers type. In this paper we determine all simply connected three dimensional Lie groups of scalar Randers type. It turns out that such groups must also admit a left invariant Riemannian metric with constant sectional curvature.

Let (nge 2) and (Omega subset mathbb {R}^n) be a bounded Lipschitz domain. Assume that (textbf{b}in L^{n*}(Omega ;mathbb {R}^n)) and (gamma ) is a non-negative function on (partial Omega ) satisfying some mild assumptions, where (n^*:=n) when (nge 3) and (n^*in (2,infty )) when (n=2). In this article, we establish the unique solvability of the Robin problems

and

in the Bessel potential space (L^p_alpha (Omega )), where (alpha in (0,2)) and (pin (1,infty )) satisfy some restraint conditions, and (varvec{nu }) denotes the outward unit normal to the boundary (partial Omega ). The results obtained in this article extend the corresponding results established by Kim and Kwon (Trans Am Math Soc 375:6537–6574, 2022) for the Dirichlet and the Neumann problems to the case of the Robin problem.

We continue the study of prime graphs of finite groups, also known as Gruenberg–Kegel graphs. The vertices of the prime graph of a finite group are the prime divisors of the group order, and two vertices p and q are connected by an edge if and only if there is an element of order pq in the group. Prime graphs of solvable groups have been characterized in graph theoretical terms only, as have been the prime graphs of groups whose only nonsolvable composition factor is (A_5). In this paper, we classify the prime graphs of all groups whose composition factors have arithmetically small orders, that is, have no more than three prime divisors in their orders. We find that all such graphs have 3-colorable complements, and we provide full characterizations of the prime graphs of such groups based on the exact type and multiplicity of the nonabelian composition factors of the group.

We introduce a capillary Chambolle-type scheme for mean curvature flow with prescribed contact angle. Our scheme includes a capillary functional instead of just the total variation. We show that the scheme is well-defined and has consistency with the energy minimizing scheme of Almgren–Taylor–Wang type. Moreover, for a planar motion in a strip, we give several examples of numerical computation of this scheme based on the split Bregman method instead of a duality method.

We study two minimization problems concerning the elastic energy on curves given by graphs subject to symmetric clamped boundary conditions. In the first, the inextensible problem, we fix the length of the curves while in the second, the extensible problem, we add a term penalizing the length. This can be considered as a one-dimensional version of the Helfrich energy. In both cases, we prove existence, uniqueness and qualitative properties of the minimizers. A key ingredient in our analysis is the use of Noether identities valid for critical points of the energy and derived from the invariance of the energy functional with respect to translations. These identities allow us also to prove curvature bounds and ordering of the minimizers even though the problem is of fourth order and hence in general does not allow for comparison principles.

Let ((lambda _k)) be a strictly increasing sequence of positive numbers such that ({sum _{k=1}^{infty } frac{1}{lambda _k} < infty }). Let f be a bounded smooth function and denote by (u= u^f) the bounded classical solution to

It is known that the following dimension-free estimate holds:

where (mu _m) is the “diagonal” Gaussian measure determined by (lambda _1, ldots , lambda _m) and (c_p > 0) is independent of f and m. This is a consequence of generalized Meyer’s inequalities [4]. We show that, if (lambda _k sim k^2), then such estimate does not hold when (p= infty ). Indeed we prove

This is in contrast to the case of (lambda _k = lambda >0), (k ge 1), where a dimension-free bound holds for (p =infty ).