### Citations

47 |
An integral inequality
- Bliss
- 1930
(Show Context)
Citation Context ...of J are solutions of differential equations of the forms(28) (r~-l]u'l ~1 sgnu')'~- Cr'-l{u] ~-1 sgnu = 0 (C = a posit ive constant)s(2) Lemma 2 and its proof are closely rela~ed to a paper of BLIss =-=[2]-=-.sGIOI~GIO TALENTI: Best constant in Sobolev inequality 365sverifying the conditions (25a, b) ands(25c) u'(r)= o(r -~"/~) as r-+O or + ~.sConversely, every solution, endowed with the properties (25), ... |

1 |
Curvature measure
- FEDRER
- 1959
(Show Context)
Citation Context ...mensional measures of the boundaries~s~(x eR": u(x) ~ t}, of the level sets of u, or with the (m-s1)-dimensional measuressof the level surfaces {x eR~: u(x)~- t}. These formulas are due to FEDEI~:Et¢ =-=[3]-=-,sFLE~I~G-I~ISH]~L [6], IJ. C. Y0V~G [13]; see also M. MIrAnDA [8]. The formula wesshall use is that of Federer, which, in our case, readss(14) fl-u( )t x= f u(x)= t}dt,s0swhere H~_I stands for (m--1)... |

1 |
FLE_~I~G, 2(ormal and integral currents
- DRER
- 1960
(Show Context)
Citation Context ...ly, it turns out thatsfunctions maximizing the ratio between the two sides of Sobolev ineqlmlity ares"characteristic functions of balls when p = 1. This is essentially a result of ~EDEI~EI~-sFLE~,I~G =-=[4]-=- and FLE~I~G-I~IsgEL [6]. Letsus describe briefly the situation.sFEDERER, FLEMING and I~ISHE]~ proved the inequality:s(5) I I" l l - l /~ ~/'(1 m/2)}lt'r~ f]Duldxsv/~m ~Jsfor every u belonging to a br... |

1 |
Functions whose partial derivatives are measures
- FLElVIIG
- 1960
(Show Context)
Citation Context ...ar. of u = f lDul axswhenever u is continuously differentiable.sBy the way, the set of all functions which are integrable in R ~ and whose totalsvariation is finite is usually called BV(Rm); see e.g. =-=[5]-=-, [7].sThe inequality (5) tell us thats(6)s{/s1~-~/~ m/2)}"sJ v/~m xtotM var. of u ,sfor every function u in B V(R~). This is obvious if u is continuously differentiable.sI fsu is not continuously dif... |

1 |
An integral ]ormula ]or total gradient variation
- FLEING
- 1960
(Show Context)
Citation Context ...nctions maximizing the ratio between the two sides of Sobolev ineqlmlity ares"characteristic functions of balls when p = 1. This is essentially a result of ~EDEI~EI~-sFLE~,I~G [4] and FLE~I~G-I~IsgEL =-=[6]-=-. Letsus describe briefly the situation.sFEDERER, FLEMING and I~ISHE]~ proved the inequality:s(5) I I" l l - l /~ ~/'(1 m/2)}lt'r~ f]Duldxsv/~m ~Jsfor every u belonging to a broad class of functions w... |

1 |
Distribuzio~i a venti derivate misure
- MIRANDA
- 1964
(Show Context)
Citation Context ...f u = f lDul axswhenever u is continuously differentiable.sBy the way, the set of all functions which are integrable in R ~ and whose totalsvariation is finite is usually called BV(Rm); see e.g. [5], =-=[7]-=-.sThe inequality (5) tell us thats(6)s{/s1~-~/~ m/2)}"sJ v/~m xtotM var. of u ,sfor every function u in B V(R~). This is obvious if u is continuously differentiable.sI fsu is not continuously differen... |

1 |
Sut minimo dell'iutegrale del gradiente li una ]unzione
- MIRANDA
- 1965
(Show Context)
Citation Context ...evel sets of u, or with the (m-s1)-dimensional measuressof the level surfaces {x eR~: u(x)~- t}. These formulas are due to FEDEI~:Et¢ [3],sFLE~I~G-I~ISH]~L [6], IJ. C. Y0V~G [13]; see also M. MIrAnDA =-=[8]-=-. The formula wesshall use is that of Federer, which, in our case, readss(14) fl-u( )t x= f u(x)= t}dt,s0swhere H~_I stands for (m--1)-dimensional (Hausdorff) measure.sversion of (14) (see [3]) iss+co... |

1 |
Disuguaglianze di Sobolev sulle ipersuper]ici minimali, l~end
- MIRADt
- 1967
(Show Context)
Citation Context ...t a connection between a Sobolev inequality and an isoperimetriesinequality for point sets can be established also if the euclidean space R ~ is replacedsby special non-flat manifolds; see M. MIrAnDA =-=[9]-=-.sOf course we have assumed in the previous discussion that m, the dimensionsof the ground space, is greater than one. In one dimension, an analogue of thesSobolev inequality is perhaps the following:... |

1 |
Minimum value ]or c in the Sobotev inequality
- ROSEN
- 1971
(Show Context)
Citation Context ...radient of usattains its maximum value C on functions u of the form (3).sWe notice thatsa discussion of the shal~p form of Sobolev inequality, restrictedsto the case m~-3 ,sp -= 27 q= 67 is in RosEr¢ =-=[10]-=-.sWe emphasize thatsour result is valid only if p ~= 1. I fsp ~ 1~ the Sobolev in-sequality behaves in a slightly different manner. In f~ct, it must be expected thatsthe ratio between a norm of u and ... |

1 | On a theorem o] ]unetionat analysis (in russian - SOBOLEV - 1938 |

1 | Applications o] junctional analysis in mathematical physics - SOBOLV - 1963 |