{
  "id": "1312.5715",
  "version": "v1",
  "published": "2013-12-19T19:53:34.000Z",
  "updated": "2013-12-19T19:53:34.000Z",
  "title": "Integral functionals on $L^p$-spaces: infima over sub-level sets",
  "authors": [
    "Biagio Ricceri"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "In this paper, we establish the following result: Let $(T,{\\cal F},\\mu)$ be a $\\sigma$-finite measure space, let $Y$ be a reflexive real Banach space, and let $\\varphi, \\psi:Y\\to {\\bf R}$ be two sequentially weakly lower semicontinuous functionals such that $$\\inf_{y\\in Y}{{\\min\\{\\varphi(y),\\psi(y)\\}}\\over {1+\\|y\\|^p}}>-\\infty$$ for some $p>0$. Moreover, assume that $\\varphi$ has no global minima, while $\\varphi+\\lambda\\psi$ is coercive and has a unique global minimum for each $\\lambda>0$. Then, for each $\\gamma\\in L^{\\infty}(T)\\cap L^1(T)\\setminus \\{0\\}$, with $\\gamma\\geq 0$, and for each $r>\\inf_{Y}\\psi$, if we put $$V_{\\gamma,r}= \\left \\{u\\in L^p(T,Y) : \\int_T\\gamma(t)\\psi(u(t))d\\mu\\leq r\\int_T\\gamma(t)d\\mu\\right \\}\\ ,$$ we have $$\\inf_{u\\in V_{\\gamma,r}} \\int_T\\gamma(t)\\varphi(u(t))d\\mu= \\inf_{\\psi^{-1}(r)}\\varphi\\int_T\\gamma(t)d\\mu\\ .$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-19T19:53:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral functionals",
      "sub-level sets",
      "reflexive real banach space",
      "unique global minimum",
      "finite measure space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.5715R"
    }
  }
}