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書評 L.Ambrosio, N.Fusco and D.Pallara: Functions of Bounded Variation and Free Discontinuity Problems

We prove that if $${t \mapsto u(t) \in BV(\mathbb{R})}$$
 is the entropy solution to a N × N strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields $$u_t + f(u)_x = 0,$$
 then up to a countable set of times $${\{t_n\}_{n \in \mathbb{N}	}}$$
 the function u(t) is in SBV, i.e. its distributional derivative u
 x
 is a measure with no Cantorian part. The proof is based on the decomposition of u
 x
 (t) into waves belonging to the characteristic families $$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal{M}(\mathbb{R}), \, \tilde r_i(t) \in \mathbb{R}^N,$$
 
and the balance of the continuous/jump part of the measures v
 i
 in regions bounded by characteristics. To this aim, a new interaction measure μ
 i,jump is introduced, controlling the creation of atoms in the measure v
 i
 (t). The main argument of the proof is that for all t where the Cantorian part of v
 i
 is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure μ
 i,jump is positive.

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension

It is well-known that solutions to the Hamilton–Jacobi equation 

$$\begin{aligned} u_{t}(t,x)+H(x,u_{x}(t,x))=0 \end{aligned}$$

fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be differentiable at a given point $(t,x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t,x)$? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t,\cdot )$ at $x$ is nonempty. Our approach uses the representation of $u$ as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem.

From pointwise to local regularity for solutions of Hamilton–Jacobi equations

We study the compactness in L 1 of the semigroup (St)t 0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of St for each t > 0 was established by P. D. Lax (1). Upper estimates for the Kolmogorov's "-entropy of the image through St of bounded sets C in L 1 \ L 1 which is denoted by

/pdf/lower-compactness-estimates-for-scalar-balance-laws-3xwfcel8ve.pdf

Lower compactness estimates for scalar balance laws

We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \[ u_t + f(u)_x = 0, \] then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. 
The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N, \] and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure $\mu_{i,\jump}$ is introduced, controlling the creation of atoms in the measure $v_i(t)$. 
The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\mu_{i,\mathrm{jump}}$ is positive.

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

In this work we study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws We assume that the source term g ∈ C1(ℝ × ℝ × ℝ+), that the flux function f ∈ C2(ℝ × ℝ × ℝ+) and that {ui ∈ ℝ : fuu(ui,x,t) = 0} is at most countable for every fixed (x,t) ∈ Ω. Our main result, which is a unification of two proposed intermediate theorems, states that BV entropy solutions of such equations belong to SBVloc(Ω). Moreover, using the theory of generalized characteristics we prove that for entropy solutions of balance laws with convex flux function, there exists a constant C > 0 such that: where C can be chosen uniformly for (x + h,t), (x,t) in any compact subset of Ω.

/pdf/sbv-regularity-of-entropy-solutions-for-a-class-of-genuinely-4mn3kbhafe.pdf

Sbv regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function

In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations $$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$$
 In particular, under the assumption that the Hamiltonian $${H\in C^2({\mathbb R}^n)}$$
 is uniformly convex, we prove that D
 x
 u and ∂
 t
 u belong to the class SBV
 loc
 (Ω).

/pdf/sbv-regularity-for-hamilton-jacobi-equations-in-mathbb-r-n-3r9lh6labw.pdf

SBV Regularity for Hamilton–Jacobi Equations in ${{\mathbb R}^n}$

We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain, as a corollary, an existence theorem for a related variational problem.

/pdf/hamilton-jacobi-equations-with-obstacles-58ohv5fori.pdf

Hamilton-Jacobi equations with obstacles

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in} \Omega\subset \mathbb R\times \mathbb R^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.

SBV regularity for Hamilton-Jacobi equations in $\R^n$

With the increasing availability of plant-based protein products that should serve as alternatives to animal-based protein products, it is necessary to develop not only environmentally friendly but also nutritious foods. Especially the protein content and quality are of concern in these products. The algorithm of NutriOpt was developed using linear programming to support the development of food products with a balanced amino acid profile while considering digestibility. The current version contains a database with 84 plant protein sources from different food groups (legumes, cereals, nuts, seeds) and with different grades of purification (flours, concentrates, isolates) from which NutriOpt can create mixtures with high protein quality while complying with constraints such as protein content, number of ingredients, and weight of the mixture. The program was tested through different case studies based on commercial plant-based drinks. It was possible to obtain formulations with a Protein Digestibility Corrected Amino Acid Score (PDCAAS) over 100 with ingredients and quantities potentially suitable for plant-based analogs. Our model can help to develop the second generation of plant-based product alternatives that can really be used as an alternative on long-term consumption. Further, there is still a great potential of expansion of the program for example to use press cakes or even to model whole menus or diets in the future.

Roger Robyr

Papers

Sbv regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function

SBV Regularity for Hamilton–Jacobi Equations in \({{\mathbb R}^n}\)

Hamilton-Jacobi equations with obstacles

SBV regularity for Hamilton-Jacobi equations in $\R^n$

Optimization of Protein Quality of Plant-Based Foods Through Digitalized Product Development