For plane domains we define a new metric close to the Poincaré metric with the Gaussian curvature k = - 4 . For this quasi-hyperbolic metric we study inequalities of isoperimetric type. It is proved that the constant of the linear quasi-hyperbolic isoperimetric inequality for admissible subdomains of a given domain is finite iff the domain does not contain the point at infinity and has a uniformly perfect boundary. In addition, we give estimates of the constants using some known numerical characteristics of domains.

In this paper, we consider parametric generalizations of a St. Venant type functional, defined on domains of the Euclidean space of dimension n≥ 2 and connected with the torsional rigidity of a domain as well as with integrals of powers of the St. Venant stress function over simply connected plane domains. We give several estimates for the generalized functionals. In particular, for bounded convex domains we obtain an essential improvement of two known results proved by R. Bañuelos, M. van den Berg, and T. Carrol (see J. London Math. Soc. 66 (2), 499–512 (2002)) and by R.G. Salahudinov (see Russian Math. (Iz. VUZ) 50 (3), 39–46 (2006)). In addition, we examine these functionals over non convex domains in two cases when a domain has uniformly perfect boundary or it is close to convex domains in a certain sense. For such a domain we prove several new estimates using power boundary moments of domains.

We consider solutions to two boundary values problems for the Poisson equation on plane domains. We prove several estimates for integrals of solutions using geometric characteristics of domains.

Hardy inequalities have numerous applications in mathematical physics and spectral theory of unbounded operators. In this paper we describe direct generalizations of integral Hardy inequalities, their improvements and analogues. We systemize the relations between various interpretations of these inequalities and describe new one-dimensional integral inequalities. We show that these known and new inequalities are valid also for complex-valued functions. We consider in details integral inequalities of Hardy, Rellich and Birman type for functions defined on bounded intervals. In particular, we provide the proofs for the generalizations and improvements of Birman integral inequalities for higher derivatives. We briefly discuss multidimensional analogues involving integrals of the powers of the modulus of the gradient of a function or of a polyharmonic operator.

This survey is devoted to a number of achievements in the theory of extremal problems in geometric function theory. The approaches to the solution of problems under consideration and the methods used are based on conformal isomorphisms and on the theory of univalent functions developed since the beginning of the 20th century. Results on integral means of conformal mappings of a disc are presented and, in particular, Dolzenko's inequality for rational functions is extended to arbitrary domains with rectifiable boundaries. Investigations in the field of Bohr-type inequalities are described. An emphasis is made on integral inequalities of Hardy and Rellich type, in which the analytic properties of inequalities are intertwined with geometric characteristics of the boundaries of domains. Results related to the solution of the Vuorinen problem on the behaviour of conformal moduli under unlimited dilations of the plane are presented. Formulae for the variation of Robin capacity are obtained. One-parameter families of rational and elliptic functions whose critical values vary in accordance with a prescribed law are characterized. The last results on Smale's conjecture and Smale's dual conjecture are described. Bibliography: 149 titles.

The Dupuit-Forchheimer (DF) approximation for unconfined groundwater flows is reduced to 2D Poisson's equation (PE), the right hand side of which involves an intensive evapotranspiration (ET) rate. A shallow water table dips inward from a constant piezometric head boundary (closed curve) such that a "dry gap," demarcated by another closed curve (unknown front), may emerge. Apriori estimates of the volume of the saturated zone and of the "dry gap" area are important for water resources management in drylands. We use the results from the theory of linear elasticity, torsion of elastic bars, for which PE is solved for the Prandtl function. Using the Poincare metrics with the Gaussian constant curvature c=-4$c\ = \ - 4$, isoperimetric inequalities are obtained for steady-state DF flows. Conformal moments and confromal radii of the domains and 2-D Hardy's type inequalities in domains, modeling the Saint Venant bar torsion problem, are involved in the obtained estimates. The studied boundary value problems (BVPs) are nonlinear for ET rates depending on the depth of the water table. In the DF model, the vadose zone (VZ) is considered as a "distributed sink" (similar to a standard "distributed source," which models recharge to the water table in humid climates). The analytical VZ is collated with one obtained from a numerical solution to BVP for Richards' equation in a 3-D saturated-unsaturated flow. BVPs for Richards' equation in cylindrical domains, solved by HYDRUS software, give the pressure head, moisture content, Darcian velocity fields, and streamlines.

Обзорная статья посвящена ряду достижений в области экстремальных проблем геометрической теории функций. В основе методов и подходов к решению рассматриваемых проблем лежат конформные изоморфизмы, а также теория однолистных функций, развивавшаяся с начала XX в. Приведены результаты по интегральным средним конформных отображений круга, в частности, дано распространение неравенства Е. П. Долженко для рациональных функций на случай произвольных областей со спрямляемыми границами. Описаны исследования в области неравенств типа Бора. Особо выделены интегральные неравенства типа Харди и Реллиха, в которых аналитические свойства неравенств тесно переплетаются с геометрическими характеристиками границ областей. Представлены результаты, касающиеся решения задачи Вуоринена о поведении конформных модулей при неограниченном растяжении плоскости. Получены формулы для вариации емкостей Робена. Охарактеризованы однопараметрические семейства рациональных и эллиптических функций, критические значения которых изменяются по заданному закону. Описаны также последние результаты по гипотезе Смейла, а также дуальной гипотезе Смейла. Библиография: 149 названий.

We justify new integral inequalities with sharp constantsfor real-valued functions vanishing on the boundary of a domain of Euclidean spaceon assuming the domain lambda-close to convex.In particular,the closure of such domain is weakly convexin the sense of Efimov–Stechkin and Vial.We describe both standard and strengthen Hardy-type inequalitieswhen instead of the gradients of test functions we usethe inner products of the gradients of the distance functionfrom a point to the boundary of the domain by test functions.To prove our main theorem,we apply several lemmas of significance in their own right.

In domains in Euclidean spaces, for test functions, we construct and prove several new Gagliardo-Nirenberg type inequalities with explicit constants. These inequalities are true in any domain, they are nonlinear, integrand functions involve the powers of the absolute values of the gradient and the Laplacian of a test function u, as well as factors of type f(vertical bar u(x)vertical bar), f'(vertical bar u(x)vertical bar), where f is a continuously differentiable non-decaying function, f(0) = 0. As weight functions, the powers of the distance from a point to the boundary of the domain serve as well as the powers of the varying hyperbolic (conformal) radius. As applications of universal inequalities of Gagliardo-Nirenberg type we obtain new integral Rellich type inequalities in planar domains with uniformly perfect boundaries. For these Rellich type L-p-inequalities we establish criteria of the positivity of the constants, obtain two-sided estimates for these constants depending on the Euclidean maximal modulus of the domain and on the parameter p >= 2. In the proof we use several scalar characteristics for domains with uniformly perfect boundaries.

Исследуются критерии конечных постоянных $C$ в серии интегральных неравенств, обобщающих неравенство Пуанкаре-Фридрихса и вариационное определение жесткости кручения области по Сен-Венану. Изопериметрическое неравенство Рэлея-Фабера-Крана и неравенство Сен-Венана-Пойа гарантируют существование конечных постоянных $C$ для областей конечного объема. Критерии существования конечной постоянной $C$ для неограниченных областей бесконечного объема известны лишь для плоских односвязных и пространственных выпуклых областей. Доказаны несколько обобщений и усилений известных результатов и получено их распространение на случай $1

We consider the Saint Venant functional P for the torsional rigidity in simply connected plane domains. We prove new lower and upper estimates of P using integrals of the conformal radius defined at any point of the domain and considered as a function. Applications of Davenport’s formula for the torsional rigidity are discussed. In addition, we present a short https://doi.org/ of the Saint Venant–Pólya isoperimetric inequality.

We study criteria for the finiteness of the constants C in integral inequalities generalizing the Poincare-Friedrichs inequality and Saint-Venant's variational definition of torsional rigidity. The Rayleigh-Faber-Krahn isoperimetric inequality and the Saint-Venant-Polya inequality guarantee the existence of finite constants C for domains of finite volume. Criteria for the existence of finite constants C for unbounded domains of infinite volume were known only in the cases of planar simply connected and spatial convex domains. We generalize and strengthen some known results and extend them to the case when 1 < p < 2. Here is one of our results. Suppose that 1 p <= 2 and Omega = Omega(0) \ K, where K subset of Omega(0) is a compact set and Omega(0) is either a planar domain with uniformly perfect boundary or a spatial domain satisfying the exterior sphere condition. Under these assumptions, a finite constant A(p-1)(Omega) exists if and only if the integral integral Omega rho(2p/(2- P)()) (x, Omega) dx is finite, where rho(x, Omega) is the distance from the point x to the boundary of Omega.

. We prove several new Hardy type inequalities in Euclidean domains; these inequalities involve the gradient of the distance function from a point to the boundary of the domain. For test functions we consider improved inequalities in form proposed by Balinsky and Evans for convex domains. Namely, in Hardy type inequalities, instead of the gradient of the test function, one takes the scalar product of the gradients of the test function and of the distance from a point to the boundary of a given domain. In the present paper, integral Hardy type inequalities are studied in non-convex 𝑛 dimensional domains having a finite inradius. We prove three new Hardy type 𝐿 𝑝 inequalities in an improved form with explicit estimates for the constants depending on the dimension of the Euclidean space 𝑛 (cid:62) 2 , the inradius of the domain and two parameters 𝑝 (cid:62) 1 , 𝑠 (cid:62) 𝑛 . Our proofs are based on three key ingredients. The first of them is related with an approximation and a special partition of the domain, in particular, we employ the approximation of the domain by subsets formed by finitely many cubes with sides parallel to the coordinate planes. The second ingredient is the representation of the domain as a countable union of sub-domains with piece-wise smooth boundaries and applying a new theorem by the author on convergence of the gradients of the distance functions for these subdomains. Moreover, we prove three new Hardy type inequalities on a finite interval, which are employed in justifying the inequalities in multi-dimensional domains.

In this paper we give a survey of selected results and open problems on integral inequalities of Mathematical Physics connected with the papers of V. Maz’ya, S. Fillippas, A. Tertikas, R. Osserman, A. Ancona, H. Brezis, M. Marcus, Y. Pinchover, E. B. Davies, A. Laptev, J. L. Fernández, J. M. Rodríguez, P. Caldiroli, R. Musina, A. A. Balinsky, W. D. Evans, R. T.  Lewis, R. G. Nasibullin, I. K. Shafigullin, the author and other mathematicians. In addition, we give some new examples and present non-linear relationships between global numerical characteristics of domains in the Euclidean space of dimension $$n\ge 2$$ .

Certain Hardy inequalities in domains of Euclidean space contain sharp but unreachable constants. V. G. Maz’ya and other authors used this fact to improve the corresponding inequalities by adding new integral terms. In this paper, a survey of results in this direction initiated by H. Brezis and M. Marcus is presented. Also, we give some generalizations of Brezis–Marcus-type inequalities to the case of Rellich-type inequalities with weights that are powers of the distance from a point to the boundary of the domain. Generalizations to the case of conformally invariant integral inequalities in simply connected and doubly connected planar hyperbolic domains are discussed.

For an open subset of the Euclidean space of dimension n we consider interior and exterior approximations by sequences of open sets. We prove convergence everywhere of the corresponding sequences of distance functions from boundary as well as convergence almost everywhere for their gradients. As applications we obtain several new Hardy-type inequalities that contain the scalar product of gradients of test functions and the gradient of the distance function from the boundary of an open subset of the Euclidean space.

Earth and Space Science Open Archive This preprint has been submitted to and is under consideration at Water Resources Research. ESSOAr is a venue for early communication or feedback before peer review. Data may be preliminary.Learn more about preprints preprintOpen AccessYou are viewing the latest version by default [v1]Saturated Water Storage in Shallow Perched Aquifer With Evapotranspiration From the Phreatic Surface and Unsaturated Lacunae: the Saint-Venant Theory RevisitedAuthorsFaritAvkhadievAnvar RKacimoviDSee all authors Farit Avkhadievobachevsky Institute of Mathematics and Mechanics, Kazan Federal Universityview email addressThe email was not providedcopy email addressAnvar R KacimoviDCorresponding Author• Submitting AuthorSultan Qaboos UniversityiDhttps://orcid.org/0000-0003-2543-3219view email addressThe email was not providedcopy email address

We prove new integral inequalities for real-valued test functions defined on subdomains of the Euclidean space. We assume that the complement of the subdomain is a non-empty convex set. We prove an extension of the Hadwiger theorems about approximations of convex compact sets by polytopes and obtain some generalizations and improvements of several Hardy type multidimensional inequalities. In particular, in the last section we present an improvement of a two-dimensional inequality, connected with the uncertainty principle of Heisenberg.

Using Stieltjes integrals we define one-parameter functionals that are monotone as a function on the parameter. We prove generalizations of some results from the papers: In contrast to these papers we prove several theorems on monotonicity of integral functionals in the case when integrating functions are not absolutely continuous. In addition, we obtain applications to isoperimetric inequalities.

We consider planar hyperbolic domains and conformally invariant functionals defined as sharp constants for Hardy type inequalities. We study relationships between these functionals and optimal constants in hyperbolic isoperimetric inequalities. The studied Hardy type inequalities involve weight functions depending on a hyperbolic radius of a domain and are conformally invariant. We prove that the positivity of Hardy constants is connected with existence of some hyperbolic isoperimetric inequalities of a special kind. We also prove a comparison theorem for Hardy constants with different numerical parameters and we study the relationships between the linear hyperbolic isoperimetric inequality in a domain and Euclidean maximum modulus of this domain. In the proofs, an essential role is played by characteristics of domains with uniformly perfect boundary. In addition, we generalize certain results from the papers J.L. Fernandez, J.M. Rodriguez, "The exponent of convergence of Riemann surfaces, bass Riemann surfaces", Ann. Acad. Sci. Fenn. Series A. I. Mathematica. 15, 165-183 (1990); V. Alvarez, D. Pestana, J.M. Rodriguez, "Isoperimetric inequalities in Riemann surfaces of infinite type", Revista Matematica Iberoamericana, 15:2, 353-425 (1999).