Hadamard Variational Formula Research Articles

The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

AbstractWe study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined by whenever is of the form , and set to otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to ‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

The Orlicz Brunn-Minkowski inequality for the first eigenvalue of p-Laplacian

In this paper, we obtain the Orlicz Hadamard variational formula and the Orlicz Brunn-Minkowski inequality for the first eigenvalue of p-Laplacian in the smooth case. Furthermore, the uniqueness of the Orlicz mixed first eigenvalue of p-Laplacian with p>1 is proved.

Some remarks on the dynamics of cell membranes

Motivated by the desire to model the entry of 1,25D into a cell by receptor mediated endocytosis, we have formulated the problem as the dynamics of a bilayer membrane. We have discussed setting the problem as a variational problem using the Helfrich modeling of the bilayer in terms of the free energy. Using a Lagrangian formulation we arrive at the Euler–Lagrange equations for the system. The model we have used depends on the amount of reagent in the neighborhood of the upper membrane. The problem thereby reduces to a moving boundary problem, which is dependent on a diffusion equation for a region changing with time. In order to solve this problem we seek the correct Neumann function for this altered. This is accomplished by deriving a Hadamard variational formula for the diffusion equation. We also offer an iterative procedure for solving this non-linear problem.

Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations

We introduce the concept of shape optimization of singular points of a weak solution in boundary value problems of partial differential equations (BVP) including fracture mechanics, shape optimization of boundary and interface, etc. Roughly speaking, singularity is a gap between weak and strong solutions. We also introduce the integral formula named the GJ-integral defined with an arbitrary subdomain as an extension of the J-integral in fracture mechanics. The GJ-integral is defined locally, is made from partial differential equations in BVP only, takes zero if the solution is regular, and expresses energy shape sensitivity. A generalization of the Hadamard variational formula is proposed and various cost functions are rewritten using the GJ-integral. In the last section there are numerical calculations using the GJ-integral with the finite element method.

Hadamard\u2019s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

The Lp Minkowski problem for q-capacity

In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q > n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q > n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

The Lp Minkowski problem for torsion

In the present paper, we introduce the so-called Lp torsion measure, Lp mixed torsion. A Hadamard variational formula for torsion under the p-sum is established. As an application, the existence of solutions to a Lp Minkowski problem for torsion is proved for any fixed p>1 and p≠n+2, which extends a result of Colesanti and Fimiani [24]. A Lp Minkowski inequality for torsion is also obtained, which implies the uniqueness of the solutions for the Lp Minkowski problem mentioned above.

A Lp Brunn-Minkowski Theory for Logarithmic Capacity

In the present paper, we first introduce the concepts of the Lp n-dimensional Logarithmic-Capacity measure and Lp mixed n-dimensional Logarithmic-Capacity and then prove some geometric properties of Lp n-dimensional Logarithmic-Capacity measure and a Lp Minkowksi inequality for the n-dimensional Logarithmic-Capacity for any fixed n ≥ 2 and p ≥ 1. As an application, we establish a Hadamard variational formula for the n-dimensional Logarithmic-Capacity under p-sum for any fixed n ≥ 2 and p ≥ 1, which extends some results of Borell (Ann. Sci. Ecole Norm. Sup. 17(3), 451–467 1984), Jerison (Adv. Math. 122(2), 262–279 1996), Colesanti-Cuoghi (Potential Anal. 22(3), 289–304 2005), Akman-Lewis-Saari-Vogel (in press) and Xiao (2013, 2018). With the Hadamard variational formula and Minkowksi inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the Logarithmic-Capacity.

The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems

In this paper, combining the p-capacity for $$p\in (1, n)$$ with the Orlicz addition of convex domains, we develop the p-capacitary Orlicz–Brunn–Minkowski theory. In particular, the Orlicz $$L_{\phi }$$ mixed p-capacity of two convex domains is introduced and its geometric interpretation is obtained by the p-capacitary Orlicz–Hadamard variational formula. The p-capacitary Orlicz–Brunn–Minkowski and Orlicz–Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The p-capacitary Orlicz–Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized p-capacitary $$L_q$$ Minkowski problems with $$q>1$$ for both discrete and general measures.

First and second Hadamard variational formulae of the Green function for general domain perturbations

The Green function of the Laplacian with the homogeneous Dirichlet boundary condition on bounded domains is considered. The variation of the Green function with respect to domain perturbations is called the Hadamard variation. In this paper, we present a unified approach to deriving the Hadamard variation. In our approach, the classical first Hadamard variation is obtained in a natural way under a less restrictive regularity assumption on the boundary smoothness. Furthermore, the second Hadamard variational formula with respective to general domain perturbations is obtained, which is an extension of the classical result of Garabedian–Schiffer in which only normal perturbation is considered.

Hadamard variational formula for the Green function of the Stokes equations under the general second order perturbation

We consider the Hadamard variational formula for the Green function of the Stokes equations with the Dirichlet boundary condition under the smooth perturbation without assuming the volume preserving property. We establish the formula for the first and the second variation for the second order perturbation.

Hadamard variational formula for eigenvalues of the Stokes equations and its application

Based on the explicit representation of the Hadamard variational formula [1] for eigenvalues of the Stokes equations, we investigate the geometry of the domain in $$\mathbb R^3$$ . It turns out that if the first variation of some eigenvalue of the Stokes equations for all volume preserving perturbations vanishes, then the domain is necessarily diffeomorphic to the 2-dimensional torus $$T^2$$ .

The Hadamard variational formula and the Minkowski problem for p-capacity

A Hadamard variational formula for p-capacity of convex bodies in Rn is established when 1<p<n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge–Ampère type equation. Uniqueness for the Minkowski problem for p-capacity is established when 1<p<n and existence and regularity when 1<p<2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p=2).

New approach to the Hadamard variational formula for the Green function of the Stokes equations

We consider the Hadamard variational formula for the Green function of the Stokes equations which describes the motion of the incompressible fluid moving infinitesimally on the bounded domain Ω with the smooth boundary ∂Ω. Under the perturbation with preserving its volume and keeping its topological type, we establish a more refined proof of its formula of the Green function not only for the first variation but also the second variation for both velocity and pressure. Our method gives a new systematic proof of the Hadamard variational formula, which enables us to deal with the higher derivatives with respect to the perturbation of domains.

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every \({\varepsilon > 0}\), we consider the Green’s matrix \({G_{\varepsilon}(x, y)}\) of the Stokes equations describing the motion of incompressible fluids in a bounded domain \({\Omega_{\varepsilon} \subset \mathbb{R}^d}\), which is a family of perturbation of domains from \({\Omega\equiv \Omega_0}\) with the smooth boundary \({\partial\Omega}\). Assuming the volume preserving property, that is, \({\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}\) for all \({\varepsilon > 0}\), we give an explicit representation formula for \({\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}\) in terms of the boundary integral on \({\partial \Omega}\) of \({G_0(x, y)}\). Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.

Hadamard variational formula for the Green function for the velocity and pressure of the Stokes equations

Hadamard variational formula for the Green function for the velocity and pressure of the Stokes equations

Exact-gradient shape optimization of a 2-D Euler flow

The optimization of an obstacle shape immersed in an Eulerian flow is investigated. In order to construct a descent method, we consider the differentiation of the flow solution with respect to the shape. In the continous case, the Hadamard variational formula yields the formal derivatives. In the discrete case, we choose an upwind method with flux splitting, and proved that an exact gradient can be derived using the adjoint state. The behavior of a gradient method is studied for a family of nozzle flows.

Hadamard's variational formula for the Szegő kernel

Hadamard's variational formula for the Szegő kernel

Hadamard's variational formula for the Bergman kernel

Hadamard's variational formula for the Bergman kernel

A remark on the Hadamard variational formula, II

A remark on the Hadamard variational formula, II