Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

We consider an optimal control problem containing a control system described by a partial nonlinear differential equation with the fractional Dirichlet–Laplacian, associated to an integral cost. We investigate the existence of optimal solutions for such a problem. In our study we use Filippov’s approach combined with a lower closure theorem for orientor fields.

Recurrence rates and form of recurrence after endoluminal ablation of the great saphenous vein – A comparison of laser ablation (EVLA) 1470 nm, radio frequency (RFITT and ClosureFast) and superheated steam with an average follow-up time of 3.8 years

Objectives In addition to combined crossectomy and stripping or pure sclerotherapy, various endovenous thermal procedures are now available for treatment, which are compared in the present study. Methods Between 2009 and 2013, the GSV was ablated in 297 patients using one of four methods: EVLA 1470 nm, ClosureFast, RFITT or superheated steam. The recurrence rate after treatment was defined as the primary endpoint. Follow-up examinations with duplex ultrasound took place 14 days, 3 months and 1 year post-operatively, and thereafter annually with average follow-up time of 3.8 years and a follow-up rate of 81 %. Results At the time of the last follow-up examination, the following complete closure rates of treated GSV were found: ClosureFast 95 %, EVLA 97 %, RFITT 79 % and superheated steam 71 %. Serious complications occurred only with superheated steam (necrosis at the puncture site). The median pain intensity recorded 14 days post-operatively was 1–3 on a scale of 1–10. Both the CIVIQ score and the VCSS were significantly improved by all endovenous thermal methods. In 5–12 % of cases, reflux was found in the previously non-refluxive AASV. Conclusions EVLA and ClosureFast are indicated for the treatment of GSV incompetence with high success rates, comparable to the results with crossectomy and stripping. The RFITT and superheated steam methods present significantly lower closure rates. Particular attention should be paid to the presence of an initially non-refluxive AASV. Since there was an increased recurrence rate over this vein, it seems reasonable to treat the AASV primarily.

Long-Term Outcome of Endorectal Advancement Flap for Complex Anorectal Fistulae

The purpose of this study was to determine the long-term outcome of endorectal advancement flap (ERF) for complex anorectal fistulae. A total of 38 ERF were performed in 36 patients (2003–2007). Mean age was 45 years. The most common fistula type was transsphincteric. Eighty-one per cent of patients had prior surgical interventions. Primary closure rate was 83 per cent. Of the six initial failures, four were noted in patients operated for recurrent rectovaginal fistula. Postoperative complications occurred in seven patients (19%). During a mean follow-up of 27 months, recurrent disease was noted in five patients (14%). All recurrences were noted in patients with left sided fistulae. At last follow-up, all patients had healed their fistula except for two. We conclude that ERF closed most complex anorectal fistulae with an acceptable complication rate and low recurrence rate. Recurrent rectovaginal fistula was associated with a lower closure rate.

A lower closure theorem for autonomous orientor fields

SynopsisGiven a set valued mapping ∑: ℝ → ∑n, we prove a closure property with respect to -convergence for the differential inclusionunder very mild assumptions on ∑.