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We prove L p (R d) × L q (R d) → L r (R d) bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the Hölder relation 1 / p + 1 / q = 1 / r . The boundedness region that we get contains at least the interior of the Hölder boundedness region of the associated single scale bilinear averaging operator. In the case of the lacunary bilinear spherical maximal function in d ≥ 2 , we prove boundedness for any p , q ∈ (1 , ∞ ] 2 , which is sharp up to boundary; we then show how to extend this result to a more degenerate family of surfaces where some curvatures are allowed to vanish. For the lacunary triangle averaging maximal operator, we have results in d ≥ 7 , and the description of the sharp region will depend on a sharp description of the Hölder bounds for the single scale triangle averaging operator, which is still open. [ABSTRACT FROM AUTHOR]

We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of the spectral gap of the graph Laplacian with standard vertex conditions: it is maximised among all graphs of fixed length by the path graph (interval), or by the loop in the restricted class of doubly connected graphs, and it is minimised among all graphs of fixed length and number of edges by the equilateral flower graph. We also establish bounds for the correctly scaled product of the spectral gap and the square of the mean distance which depend only on combinatorial, and not metric, features of the graph. This raises the open question whether this product admits absolute upper and lower bounds valid on all compact metric graphs. [ABSTRACT FROM AUTHOR]

We study the metric properties of level surfaces for classes of smooth noncontact mappings from arbitrary Carnot groups into two-step ones with some constraints on the dimensions of horizontal subbundles and the subbundles corresponding to degree 2 fields. We calculate the Hausdorff dimension of the level surfaces with respect to the sub-Riemannian quasimetric and derive an analytical relation between the Hausdorff measures for the sub-Riemannian quasimetric and the Riemannian metric. As application, we establish a new form of coarea formula, also proving that the new coarea factor is well defined. [ABSTRACT FROM AUTHOR]

The coarea formula is of crucial importance in analysis and has numerous applications in a variety of deep problems. Here we give a modern proof and discuss some of the applications. [ABSTRACT FROM AUTHOR]

We consider a model example of noncontact mappings of Heisenberg groups where the dimension of the source space is greater than the dimension of the target space. We derive metric properties of level surfaces and prove an analog of the coarea formula. [ABSTRACT FROM AUTHOR]

8 V. Caselles, On the entropy conditions for some flux limited diffusion equations, J. Differential Equations 250 (2011), no. 8, 3311-3348. 18 (2018), no. 1, 1-28. 30 G. P. Leonardi and G. Saracco, The prescribed mean curvature equation in weakly regular domains, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, Paper No. 9. 31 G. P. Leonardi and G. Saracco, Rigidity and trace properties of divergence-measure vector fields, Adv. Calc. Keywords: Divergence-measure vector fields; functions of bounded variation; coarea formula; Gauss-Green formula; semicontinuity; 26B30; 49Q15; 49J45 EN Divergence-measure vector fields functions of bounded variation coarea formula Gauss-Green formula semicontinuity 26B30 49Q15 49J45 787 810 24 10/03/22 20221001 NES 221001 1 cations, Ann. [Extracted from the article]

We consider submanifolds of sub-Riemannian Carnot groups with intrinsic C 1 regularity ( C H 1 ). Our first main result is an area formula for C H 1 intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing C H 1 submanifolds into level sets of a C H 1 function. [ABSTRACT FROM AUTHOR]

Considering the class of contact mappings of Carnot groups with a multidimensional sub-Lorentzian structure on the preimages, we prove that the tangent plane approximates the level sets to a higher order than in the classical case. We also obtain a coarea formula for such mappings with a sub-Lorentzian measure on the level sets. [ABSTRACT FROM AUTHOR]

We establish the coarea formula as an expression for the measure of a subset of a Carnot group in terms of the sub-Lorentzian measure of the intersections of the subset with the level sets of a vector function. We describe the conditions for the level sets of vector functions to be spacelike and find the metric characteristics of these surfaces. Also, we address a series of relevant questions, in particular, about the uniqueness of the coarea factor. [ABSTRACT FROM AUTHOR]

We consider C1-smooth vector functions defined on Carnot groups of arbitrary depth, deduce conditions for space-likeness of their level surfaces, and describe their metric properties from the viewpoint of sub-Lorentzian geometry. We prove the coarea formula as an expression of the measure of a subset of a Carnot group in terms of the sub-Lorentzian measures of its intersections with level sets of a vector function. [ABSTRACT FROM AUTHOR]

We obtain the metric properties of level sets of certain sufficiently smooth mappings of two-step Carnot groups which are Hölder in the sub-Riemannian sense. In particular, we calculate the Hausdorff dimension of these sets and prove a coarea formula of a new type. [ABSTRACT FROM AUTHOR]

This paper introduces a novel boundary integral approach of shape uncertainty quantification for the Helmholtz scattering problem in the framework of the so-called parametric method. The key idea is to construct an integration grid whose associated weight function encompasses the irregularities and non-smoothness imposed by the random boundary. Thus, the solution can be evaluated accurately with relatively low number of grid points. The integration grid is obtained by employing a low-dimensional spatial embedding using the coarea formula. The proposed method can handle large variation as well as non-smoothness of the random boundary. For the ease of presentation the theory is restricted to star-shaped obstacles in low-dimensional setting. Higher spatial and parametric dimensional cases are discussed, though, not extensively explored in the current study. [ABSTRACT FROM AUTHOR]

Abstract In this paper we obtain a very general Gauss–Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing (A , D u) when A is a bounded divergence measure vector field and u is a bounded function of bounded variation. [ABSTRACT FROM AUTHOR]

We consider C1-functions defined on two-step Carnot groups with a sub-Lorentzian structure defined by one horizontal direction with a negative squared length along it, and prove a nonholonomic coarea formula. A result of interest in itself concerns the correctness of the problem statement, namely, the level sets have to be spacelike. [ABSTRACT FROM AUTHOR]

We present a piecewise Korn inequality for generalized special functions of bounded deformation (GSBD2) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that GSBD2 functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in GSBD2 in dimension two. Moreover, for all d ≥ 2 we show GSBD2 (Ω) ⊂ (GBV (Ω; R))d and the embedding SBD2 (Ω) ∩ L∞(Ω; Rd) ,→ SBV (Ω; Rd) into the space of special functions of bounded variation (SBV ). Finally, we present a Korn-Poincaré inequality for functions with small jump sets in arbitrary space dimension. [ABSTRACT FROM AUTHOR]

Swimmers caught in a rip current flowing away from the shore are advised to swim orthogonally to the current to escape it. We describe a mathematical principle in a similar spirit. More precisely, we consider flows γ in the plane induced by incompressible vector fields v:R2→R2 satisfying c1<‖v‖ The length ℓ a flow curve γ˙(t)=v(γ(t)) until γ leaves a disk of radius 1 centered at the initial position can be as long as ℓ∼c2/c1. The same is true for the orthogonal flow v⊥=(-v2,v1). We show that a combination does strictly better: there always exists a curve flowing first along v⊥ and then along v which escapes the unit disk before reaching the length 4πc2/c1. Moreover, if the escape length of v is uniformly ∼c2/c1, then the escape length of v⊥ is uniformly ∼1 (allowing for a fast escape from the current). We also prove an elementary quantitative Poincaré-Bendixson theorem that seems to be new. [ABSTRACT FROM AUTHOR]

In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into R) is due to Kronrod (Uspechi Matem Nauk 5(1):24-134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418-491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477-492, 2002), Fleming and Rishel (Arch Math 11(1):218-222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31-91, 1988), is not valid when applied to C∞ functions. The gap appears only for the non generic set of non Morse functions. [ABSTRACT FROM AUTHOR]

We introduce a novel method to compute approximations of integrals over implicitly defined hypersurfaces. The new method is based on a weak formulation in L²(0; 1) that uses the coarea formula to circumvent an explicit integration over the hypersurfaces. As such it is possible to use standard quadrature rules in the spirit of hp/spectral finite element methods, and the expensive computation of explicit hypersurface parametrizations is avoided. We derive error estimates showing that high order convergence can be achieved provided the integrand and the hypersurface defining function are suffciently smooth. The theoretical results are supplemented by numerical experiments including an application for plasma modeling in nuclear fusion. [ABSTRACT FROM AUTHOR]

We propose a new definition for the gradient field of a discrete image defined on a twice finer grid. The differentiation process from an image to its gradient field is viewed as the inverse operation of linear integration, and the proposed mapping is nonlinear. Then, we define the total variation of an image as the ℓ1 norm of its gradient field amplitude. This new definition of the total variation yields sharp edges and has better isotropy than the classical definition. [ABSTRACT FROM AUTHOR]

We prove a new vectorial functional inequality of Poincaré–Beckner type. The inequality may be interpreted as an entropy–entropy production one for a gradient flow in the metric space of Radon measures. The proof uses subtle analysis of combinations of related super- and sub-level sets employing the coarea formula and the relative isoperimetric inequality. [ABSTRACT FROM AUTHOR]

Let X = X [ a , b ] = { x : a < | x | < b } ⊂ R n with 0 < a < b < ∞ fixed be an open annulus and consider the energy functional, F [ u ; X ] = 1 2 ∫ X | ∇ u | 2 | u | 2 d x , over the space of admissible incompressible Sobolev maps A ϕ ( X ) = { u ∈ W 1 , 2 ( X , R n ) : det ∇ u = 1 a.e. in X and u | ∂ X = ϕ } , where ϕ is the identity map of X ¯ . Motivated by the earlier works (Taheri (2005), (2009)) in this paper we examine the twist maps as extremisers of F over A ϕ ( X ) and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case n = 2 where A ϕ ( X ) is a union of infinitely many disjoint homotopy classes we establish the minimality of these extremising twists in their respective homotopy classes a result that then leads to the latter twists being L 1 -local minimisers of F in A ϕ ( X ) . We discuss variants and extensions to higher dimensions as well as to related energy functionals. [ABSTRACT FROM AUTHOR]

The Morse–Sard theorem requires that a mapping v : R n → R m is of class C k , k > max ⁡ ( n − m , 0 ) . In 1957 Dubovitskiĭ generalized this result by proving that almost all level sets for a C k mapping have H s -negligible intersection with its critical set, where s = max ⁡ ( n − m − k + 1 , 0 ) . Here the critical set, or m -critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) < m } . Another generalization was obtained independently by Dubovitskiĭ and Federer in 1966, namely for C k mappings v : R n → R d and integers m ≤ d they proved that the set of m -critical values v ( Z v , m ) is H q ∘ -negligible for q ∘ = m − 1 + n − m + 1 k . They also established the sharpness of these results within the C k category. Here we prove that Dubovitskiĭ's theorem can be generalized to the case of continuous mappings of the Sobolev–Lorentz class W p , 1 k ( R n , R d ) , p = n k (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin N -property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a bridge theorem that includes all the above results as particular cases. As a limiting case in this bridge theorem we also establish a new coarea type formula: if E ⊂ { x ∈ R n : rank ∇ v ( x ) ≤ m } , then ∫ E J m v ( x ) d x = ∫ R d H n − m ( E ∩ v − 1 ( y ) ) d H m ( y ) . The mapping v is R d -valued, with arbitrary d , and the formula is obtained without any restrictions on the image v ( R n ) (such as m -rectifiability or σ -finiteness with respect to the m -Hausdorff measure). These last results are new also for smooth mappings, but are presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J. Bourgain (2013, 2015). [ABSTRACT FROM AUTHOR]