다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf

User’s guide to viscosity solutions of second order partial differential equations

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

I Preliminaries.- 1. Metric properties of normed spaces.- 1.1 Duality mappings.- 1.2 Strictly convex normed spaces.- 1.3 Uniformly convex Banach spaces.- 2. Vectorial functions defined on real intervals.- 2.1 Absolutely continuous vectorial functions.- 2.2 Vectorial distributions and Wk,p spaces.- 2.3 Sobolev spaces.- 3. Semigroups of continuous linear operators.- 3.1 Semigroups of class (C0). Hille-Yosida theorem.- 3.2 Analytic semigroups.- 3.3 Nonhomogeneous linear differential equations.- II Nonlinear Operators in Banach Spaces.- 1. Maximal monotone operators.- 1.1 Definitions and fundamental concepts.- 1.2 A general perturbation theorem.- 1.3 A nonlinear elliptic boundary problem.- 2. Subdifferential mappings.- 2.1 Lower semicontinuous convex functions.- 2.2 Subdifferentials of convex functions.- 2.3 Some examples of cyclically monotone operators.- 3. Dissipative sets in Banach spaces.- 3.1 Basic properties of dissipative sets.- 3.2 Perturbations of dissipative sets.- 3.3 Riccati equations in Hilbert spaces.- Bibliographical notes.- III Differential Equations in Banach Spaces.- 1. Semigroups of nonlinear contractions in Banach spaces.- 1.1 General properties of nonlinear semigroups.- 1.2 The exponential formula.- 1.3 Convergence theorems.- 1.4 Generation of nonlinear semigroups.- 2. Quasi-autonomous differential equations.- 2.1 Existence theorems.- 2.2 Periodic solutions.- 2.3 Examples.- 3. Differential equations associated with continuous dissipative operators.- 3.1 A general existence result.- 3.2 Continuous perturbations of m-dissipative operators.- 3.3 Semi-linear second-order elliptic equations in L1.- 4. Time-dependent nonlinear differential equations.- 4.1 Evolution equations associated with dissipative sets.- 4.2 Evolution equations associated with nonlinear monotone hemicon-tinuous operators.- Bibliographical notes.- IV Nonlinear Differential Equations in Hilbert Spaces.- 1. Nonlinear semigroups in Hilbert spaces.- 1.1 Nonlinear version of the Hille-Yosida theorem.- 1.2 Exponential formulae.- 1.3 Invariant sets with respect to nonlinear semigroups.- 2. Smoothing effect on initial data.- 2.1 The case in which A = ? ?.- 2.2 The case in which int D(A) ? ?.- 2.3 Applications.- 3. Variational evolution inequations.- 3.1 Unilateral conditions on u(t).- 3.2 Unilateral conditions on $$ \frac{{du}}{{dt}}(t) $$.- 3.3 A class of nonlinear variational inequations.- 3.4 Applications.- 4. Nonlinear Volterra equations with positive kernels in Hilbert spaces.- 4.1 Positive kernels.- 4.2 Equation (4.1) with A = ? ?.- 4.3 Equation (4.1) with A demicontinuous.- 4.4 A class of integro-differential equations.- 4.5 Further investigation of the preceding case.- Bibliographical notes.- V Second Order Nonlinear Differential Equations.- 1. Nonlinear differential equations of hyperbolic type.- 1.1 The equation $$ \frac{{{d^2}u}}{{d{t^2}}} + Au + M\left( {\frac{{du}}{{dt}}} \right) \mathrel\backepsilon f $$.- 1.2 Further investigation of the preceding case.- 1.3 Examples.- 1.4 Singular perturbations and hyperbolic variational inequations.- 1.5 Nonlinear wave equation.- 2. Boundary value problems for second order nonlinear differential equations.- 2.1 A class of two-point boundary value problems.- 2.2 Examples.- 2.3 A boundary value problem on half-axis.- 2.4 The square root of a nonlinear maximal monotone operator.- Bibliographical notes.

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Nonlinear semigroups and differential equations in Banach spaces

Fundamentals of Functional Analysis.- Convex Functions.- Convex Programming.- Convex Control Problems in Banach Spaces.

Convexity and optimization in Banach spaces

Fundamental Functional Analysis.- Maximal Monotone Operators in Banach Spaces.- Accretive Nonlinear Operators in Banach Spaces.- The Cauchy Problem in Banach Spaces.- Existence Theory of Nonlinear Dissipative Dynamics.

Nonlinear Differential Equations of Monotone Types in Banach Spaces

Nonlinear operators of monotone type controlled elliptical variational inequalities nonlinear accretive differential equations optimal control of parabolic variational inequalities optimal control in real time.

Viorel Barbu

Papers

Nonlinear semigroups and differential equations in Banach spaces

Convexity and optimization in Banach spaces

Nonlinear Differential Equations of Monotone Types in Banach Spaces

Analysis and control of nonlinear infinite dimensional systems

Optimal control of variational inequalities