Abstract
A smooth linear complex partial differential equation in two variables which is without solutions is found.
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References
Lewy, H., An example of a smooth linear partial differential equation without solution, Ann. Math., 1957, 66:155–158.
Treves, F., The equation \(\left[ {\frac{1}{4}\left( {\frac{{\partial ^2 }}{{\partial x^2 }} + \frac{{\partial ^2 }}{{\partial y^2 }}} \right) + (x^2 + y^2 )\frac{{\partial ^2 }}{{\partial t^2 }}\left( {x\frac{\partial }{{\partial y}} + y\frac{\partial }{{\partial x}}} \right)\frac{{\partial ^2 }}{{\partial t}}} \right]^2 u + \frac{{\partial ^2 }}{{\partial t^2 }}u = f\) with real coefficients, is “without solutions”, Bull. Amer. Math. Soc., 1962, 68:332.
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Supported by the NNSF of China (19631050, 19771072) and the NSF of Zhejiang Province (195026).
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Baojun, B., Junjie, L. A remark on the smooth linear partial differential equations in two variables without solutions. Appl. Math.- J. Chin. Univ. 16, 8–10 (2001). https://doi.org/10.1007/s11766-001-0031-1
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DOI: https://doi.org/10.1007/s11766-001-0031-1