Loss of derivatives

We give a proof of the Nirenberg-Treves conjecture: that local solvability of principal-type pseudo-differential operators is equivalent to condition (Ψ). This condition rules out sign changes from − to + of the imaginary part of the principal symbol along the oriented bicharacteristics of the real part. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case).

41p noel_noel 17-01-2013 44 7   Download

Dedicated to Yum-Tong Siu for his 60th birthday. Abstract Let {X1 , . . . , Xp } be complex-valued vector fields in Rn and assume that they satisfy the bracket condition (i.e. that their Lie algebra spans all vector fields). Our object is to study the operator E = Xi∗ Xi , where Xi∗ is the L2 adjoint of Xi . A result of H¨rmander is that when the Xi are real then E is o hypoelliptic and furthemore it is subelliptic (the restriction of a destribution u to an open set U is “smoother” then the restriction...

45p noel_noel 17-01-2013 47 7   Download

The project is to transfer a small suite of laboratory technologies to develop a capability in Vietnam for the diagnosis and control of the major disease of pigs, causing financial loss to both large and small scale farmers. The project inputs do not finish at that point, but further include the provision of training to ensure that the benefits to be derived from the enhanced technological capability will be available to farmers. Hence the project spans a number of activities from the laboratory to the farm...

16p tam_xuan 02-03-2012 63 5   Download