Variational structure of the vn2-Yamabe problem

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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Mathematical modeling of variational structure of a torque moment measurement

Construction of Oil and Gas Wells on Land and Sea ◽

10.30713/0130-3872-2018-6-11-13 ◽

2018 ◽

pp. 11-13

Author(s):

B.A. Perminov ◽

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V.B. Perminov ◽

Z.Kh. Yagubov ◽

A.S. Rozanov ◽

...

Keyword(s):

Mathematical Modeling ◽

Moment Measurement ◽

Variational Structure ◽

Torque Moment

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Learned Collaborative Stereo Refinement

International Journal of Computer Vision ◽

10.1007/s11263-021-01485-5 ◽

2021 ◽

Author(s):

Patrick Knöbelreiter ◽

Thomas Pock

Keyword(s):

Structural Properties ◽

Gradient Method ◽

Color Image ◽

Image Space ◽

Activation Functions ◽

Disparity Map ◽

Proximal Gradient Method ◽

Variational Structure ◽

Disparity Maps ◽

The Individual

AbstractIn this work, we propose a learning-based method to denoise and refine disparity maps. The proposed variational network arises naturally from unrolling the iterates of a proximal gradient method applied to a variational energy defined in a joint disparity, color, and confidence image space. Our method allows to learn a robust collaborative regularizer leveraging the joint statistics of the color image, the confidence map and the disparity map. Due to the variational structure of our method, the individual steps can be easily visualized, thus enabling interpretability of the method. We can therefore provide interesting insights into how our method refines and denoises disparity maps. To this end, we can visualize and interpret the learned filters and activation functions and prove the increased reliability of the predicted pixel-wise confidence maps. Furthermore, the optimization based structure of our refinement module allows us to compute eigen disparity maps, which reveal structural properties of our refinement module. The efficiency of our method is demonstrated on the publicly available stereo benchmarks Middlebury 2014 and Kitti 2015.

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A mass-conserved tumor invasion system with quasi-variational degenerate diffusion

Analysis and Applications ◽

10.1142/s0219530521500159 ◽

2021 ◽

pp. 1-66

Author(s):

Akio Ito

Keyword(s):

Hilbert Space ◽

Tumor Cells ◽

Tumor Invasion ◽

Real Hilbert Space ◽

Initial Boundary ◽

Inner Product ◽

Boundary Problems ◽

Theory Of Evolution ◽

Cross Diffusion ◽

Variational Structure

This paper deals with a nonlinear system (S) composed of three PDEs and one ODE below: [Formula: see text] The system (S) was proposed as one of the mathematical models which describe tumor invasion phenomena with chemotaxis effects. The most important and interesting point is that the diffusion coefficient of tumor cells, denoted by [Formula: see text], is influenced by both nonlocal effect of a chemical attractive substance, denoted by [Formula: see text], and the local one of extracellular matrix, denoted by [Formula: see text]. From this point, the first PDE in (S) contains a nonlinear cross diffusion. Actually, this mathematical setting gives an inner product of a suitable real Hilbert space, which governs the dynamics of the density of tumor cells [Formula: see text], a quasi-variational structure. Hence, the first purpose in this paper is to make it clear what this real Hilbert space is. After this, we show the existence of strong time local solutions to the initial-boundary problems associated with (S) when the space dimension is [Formula: see text] by applying the general theory of evolution inclusions on real Hilbert spaces with quasi-variational structures. Moreover, for the case [Formula: see text] we succeed in constructing a strong time global solution.

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