Unique Continuation Properties and Quantitative Estimates of Unique Continuation for Parabolic Equations

2009 ◽  
pp. 421-500 ◽  
Author(s):  
Sergio Vessella
Keyword(s):  
Parabolic Equations ◽  
Unique Continuation ◽  
Quantitative Estimates

The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian

2019 ◽  
Vol 25 ◽  
pp. 7 ◽  
Author(s):  
Xin Yu ◽  
Liang Zhang
Keyword(s):  
Optimal Control ◽  
Bounded Domain ◽  
Parabolic Equations ◽  
Fractional Laplacian ◽  
Optimal Control Problems ◽  
Unique Continuation ◽  
Control Problems ◽  
Time Varying ◽  
Observability Inequality ◽  
Quantitative Unique Continuation

In this paper, we establish the bang-bang property of time and norm optimal control problems for parabolic equations governed by time-varying fractional Laplacian, evolved in a bounded domain of ℝd. We firstly get a quantitative unique continuation at one point in time for parabolic equations governed by time-varying fractional Laplacian. Then, we establish an observability inequality from measurable sets in time for solutions of the above-mentioned equations. Finally, with the aid of the observability inequality, the bang-bang property of time and norm optimal control problems can be obtained.

scholarly journals Unique Continuation Property of Non-Positive Weak Subsolutions for Parabolic Equations of Higher Order

1964 ◽  
Vol 24 ◽  
pp. 241-248
Author(s):  
Kazunari Hayashida
Keyword(s):  
Maximum Principle ◽  
Differential Operator ◽  
Parabolic Equations ◽  
Unique Continuation ◽  
Second Order ◽  
Wide Sense ◽  
The Maximum Principle ◽  
Unique Continuation Property ◽  
Continuation Property ◽  
Order Continuous

When L is a parabolic differential operator of second order, Nirenberg [6] proved the maximum principle for the function u which has second order continuous derivatives and satisfies Lu≧0. Recently Friedman [2] has proved the maximum principle for the measurable function satisfying Lu≧O in the wide sense. This function is named a weakly L-subparabolic function. On the other hand, Littman [5] earlier than Friedman, has defined a weakly A- subharmonic function for an elliptic differential operator A of second order and has showed the maximum principle for it.

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