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Evolution equations with lack of convexity. (English) Zbl 0545.46029
Let $$\Omega$$ be an open subset of a real Hilbert space H, whose norm and scalar product are denoted by $$| \cdot |$$ and ($$\cdot | \cdot)$$. If $$f:\quad \Omega \to {\mathbb{R}}\cup \{+\infty \}$$ is a function, set $\partial^-f(u)=\{\alpha \in H:\lim \quad \inf_{v\to u}\frac{f(v)-f(u)-(\alpha | v-u)}{| v-u|}\geq 0\},\quad if\quad f(u)<+\infty;$
$\partial^-f(u)=\emptyset,\quad if\quad f(u)=+\infty.$ If f is lower semicontinuous (with respect to the norm topology), f is said to have a $$\phi$$-monotone subdifferential, if there exists a continuous function $$\phi:\quad \Omega \times {\mathbb{R}}^ 2\to {\mathbb{R}}^+$$ such that $(\alpha -\beta | u-v)\geq - [\phi(u,f(u),| \alpha |)+\phi(v,f(v),| \beta |)]\quad | u-v|^ 2$ whenever $$\partial^-f(u)\neq \emptyset$$, $$\partial^-f(v)\neq \emptyset$$, $$\alpha \in \partial^-f(u)$$, $$\beta \in \partial^-f(v)$$. In this paper some general properties of this class of functions are studied and some theorems of existence, uniqueness, regularity and convergence, concerning the associated evolution equation $$U'(t)\in -\partial^-f(U(t))$$ are proved.

##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces 46A50 Compactness in topological linear spaces; angelic spaces, etc. 58D25 Equations in function spaces; evolution equations 49J27 Existence theories for problems in abstract spaces
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##### References:
 [1] Aubin, J.P., Mathematical methods of game and economic theory, (1979), North-Holland Amsterdam · Zbl 0452.90093 [2] Brezis, H., Opérateurs maximaux monotones, Vol. 50, (1973), North-Holland Amsterdam, Notes de Mathematica [3] Brøndsted, A.; Rockafellar, R.T., On the subdifferentiability of convex functions, Proc. am. math. soc., 16, 605-611, (1965) · Zbl 0141.11801 [4] Clarke, F.H., Optimization and non-smooth analysis, (1983), John Wiley [5] Crandall, M.G.; Liggett, T.M., Generation of semi-groups of nonlinear transformations on general Banach spaces, Am. J. math., 93, 265-298, (1971) · Zbl 0226.47038 [6] De Giorgi, E., Generalized limits in calculus of variations, () · Zbl 0493.49004 [7] De Giorgi, E.; Degiovanni, M.; Marino, A.; Tosques, M., Evolution equations for a class of non-linear operators, Atti accad. naz. lincei, 75, 1-8, (1983) · Zbl 0597.47045 [8] De Giorgi, E.; Degiovanni, M.; Tosques, M., Recenti sviluppi Della γ-convergenza in problemi ellittici, parabolici ed iperbolici, () [9] De Giorgi, E.; Marino, A.; Tosques, M., Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti accad. naz. lincei, 68, 180-187, (1980) · Zbl 0465.47041 [10] De Giorgi, E.; Marino, A.; Tosques, M., Funzioni (p, q)-convesse, Atti accad. naz. lincei, 73, 6-14, (1982) · Zbl 0521.49011 [11] Degiovanni, M.; Marino, A.; Tosques, M., General properties of (p, q)-convex functions and (p, q)-monotone operators, Ric. mat. (Naples), 32, 285-319, (1983) · Zbl 0555.49007 [12] Degiovanni, M.; Marino, A.; Tosques, M., Evolution equations associated with (p, q)-convex functions and (p, q)-monotone operators, Ric. mat. (Naples), 33, 81-112, (1984) · Zbl 0582.49005 [13] Degiovanni, M.; Marino, A.; Tosques, M., Critical points and evolution equations, (), 184-192 [14] Edelstein, M., On nearest points of sets in uniformly convex Banach spaces, J. lond. math. soc., 43, 375-377, (1968) · Zbl 0183.40403 [15] Ekeland, I., Nonconvex minimization problems, Bull. am. math. soc., 1, 443-474, (1979) · Zbl 0441.49011 [16] Marino, A.; Scolozzi, D., Geodetiche con ostacolo, Boll. un. mat. ital., 2-B, 1-31, (1983) · Zbl 0563.53034 [17] Marino, A.; Scolozzi, D., Punti inferiormente stazionari ed equazioni di evoluzione con vincoli unilaterali non convessi, Rc. sem. mat. fis. milano, 52, 393-414, (1982) · Zbl 0567.35005 [18] Marino, A.; Scolozzi, D., Autovalori dell’operatore di Laplace ed equazioni di evoluzione in presence di ostacolo, () [19] Marino, A.; Tosques, M., Curves of maximal slope for a certain class of non regular functions, Boll. un. mat. ital., 1-B, 143-170, (1982) · Zbl 0495.58012 [20] Marino A. & Tosques M., Existence and properties of the curves of maximal slope (to appear). · Zbl 0495.58012 [21] Pazy, A., Semigroups of non-linear contraction in Hilbert space, () · Zbl 0399.47057 [22] Rockafellar, R.T., Generalized directional derivatives and subgradients of non convex functions, Can. J. math., 32, 257-280, (1980) · Zbl 0447.49009
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