Design of cavitation-free hydrofoils by a given pressure envelope

In this paper we shortly describe basic aspects of the theory of pressure envelopes which in the frame work of potential flows allows one to design a wing section shape that generates exactly a specified pressure envelope. By means of this theory we analyze and modify a series of hydrofoils designed by Eppler. The modifications based on shifts and proportional stretches of the dependence of the maximum velocity on the angle of attack. Besides, applying the theory, we solve an optimal problem and design a series of optimal hydrofoils which have a maximal width of the pressure bucket. We present accurate estimates of the maximal width as a function of the cavitation number and angle of attack.http://deepblue.lib.umich.edu/bitstream/2027.42/84220/1/CAV2009-final175.pd

Sufficient conditions for the univalence of quasiconformal mappings

Using two methods, quasiconformal continuation involving a theorem of Hadamard and direct estimation of f(z2)-f(z1), we obtain sufficient conditions for the univalence of continuously differentiable mappings f(z) of plane domains which, in the case of conformal mappings, reduce to both well-known and new results. © 1976 Plenum Publishing Corporation

Estimates of Hardy-Rellich constants for polyharmonic operators and their generalizations

© Avkhadiev F.G. 2017. We prove the lower bounds for the functions introduced as the maximal constants in the Hardy and Rellich type inequalities for polyharmonic operator of order m in domains in a Euclidean space. In the proofs we employ essentially the known integral inequality by O.A. Ladyzhenskaya and its generalizations. For the convex domains we establish two generalizations of the known results obtained in the paper M.P. Owen, Proc. Royal Soc. Edinburgh, 1999 and in the book A.A. Balinsky, W.D. Evans, R.T. Lewis, The analysis and geometry of hardy's inequality, Springer, 2015. In particular, we obtain a new proof of the theorem by M.P. Owen for polyharmonic operators in convex domains. For the case of arbitrary domains we prove universal lower bound for the constants in the inequalities for mth order polyharmonic operators by using the products of m different constants in Hardy type inequalities. This allows us to obtain explicit lower bounds for the constants in Rellich type inequalities for the dimension two and three. In the last section of the paper we discuss two open problems. One of them is similar to the problem by E.B. Davies on the upper bounds for the Hardy constants. The other problem concerns the comparison of the constants in Hardy and Rellich type inequalities for the operators defined in three-dimensional domains

Families of domains with best possible hardy constant

We geometrically describe families of non-convex plane and spatial domains in which the basic Hardy inequality is valid with the constant 1/4. In our constructions we use some new constants depending on the dimension; we determine them as roots of Lamb-type equations. We also use the constant defined by E. B. Davies. © 2013 Allerton Press, Inc

Hardy type L p-inequalities in r-close-to-convex domains

© 2015, Allerton Press, Inc. We describe non-convex domains for which the Hardy constants are the same as for convex domains

Sharp estimates for functions with a pole and logarithmic singularity

We consider functions with a pole and a logarithmic singularity. We obtain sharp estimates for the Schwarzian and the Taylor coefficients of the holomorphic part of such functions. We also describe geometric properties of conformal mappings of the exterior of the unit disc with a cut that connects some boundary point with the point at infinity. © 2011 Allerton Press, Inc

Rellich Type Inequalities with Weights in Plane Domains

© 2018, Pleiades Publishing, Ltd. We determine some special functionals as sharp constants in integral inequalities for test functions, defined on plane domains. First we prove a new one dimensional integral inequality. Also, we prove some generalizations of a classical Rellich result for two dimensional case, when there is an additional restriction for Fourier coefficients of the test functions. In addition, we examine a Rellich type inequality in plane domains with infinite Euclidean maximal modulus. As an application of our results we present a new simple proof of a remarkable theorem of P. Caldiroli and R. Musina from their paper “Rellich inequalities with weights”, published in Calc. Var. 45 (2012), 147–164

Hardy type inequalities in higher dimensions with explicit estimate of constants

Let Ω be an open set in ℝn such that Ω ≠ ℝn. For 1 ≤ p n, then for arbitrary open sets Ω ⊂ ℝn (Ω ≠ ℝn) and any p ∈ [1, ∞) the sharp inequality cp(s, Ω) ≤ p/(s - n) is valid. This gives a solution of a known problem due to J.L.Lewis [31] and A.Wannebo [44]. Estimates of constants in certain other Hardy and Rellich type inequalities are also considered. In particular, we obtain an improved version of a Hardy type inequality by H.Brezis and M.Marcus [13] for convex domains and give its generalizations

Sharp constants in hardy type inequalities

© 2015, Allerton Press, Inc. We prove new weighted Hardy type inequalities with sharp constants and describe their applications to inequalities in multidimensional domains

A geometric description of domains whose Hardy constant is equal to 1/4

© 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd. We give a geometric description of families of non-convex planar and spatial domains in which the following Hardy inequality holds: the Dirichlet integral of any smooth compactly supported function f on the domain is greater than or equal to one quarter of the integral of f2(x)/δ2(x), where δ(x) is the distance from x to the boundary of the domain. Our geometric description is based analytically on new one-dimensional Hardy-type inequalities with special weights and on new constants related to these inequalities and hypergeometric functions